Recent zbMATH articles in MSC 49https://zbmath.org/atom/cc/492024-02-28T19:32:02.718555ZUnknown authorWerkzeugEkeland variational principle in complete weakly symmetric \((1, q_2)\)-quasimetric spaces and applicationshttps://zbmath.org/1527.260072024-02-28T19:32:02.718555Z"Yang, Hongzhen"https://zbmath.org/authors/?q=ai:yang.hongzhen"Li, Jun"https://zbmath.org/authors/?q=ai:li.jun.15|li.jun.12|li.jun.19|li.jun|li.jun.11|li.jun.6|li.jun.7|li.jun.9|li.jun.52|li.jun.8|li.jun.16|li.jun.18|li.jun.24|li.jun.1|li.jun.21|li.jun.3|li.jun.23The Ekeland variational principle is extended for bifunctions in complete weakly symmetric \((1, q_2)\)-quasimetric spaces and it turns out to be equivalent to the Oettli-Théra theorem, Caristi-Kirk fixed point theorem and Takahashi's nonconvex minimization principle. A version of the Ekeland variational principle for systems of bifunctions is derived, too. These lead to guaranteeing the existence of solutions to quasi-equilibrium problems under the compactness condition and to uncountable systems of quasi-equilibrium problems under the generalized Caristi-like condition.
Reviewer: Sorin-Mihai Grad (Paris)Set-valued functions of bounded generalized variation and set-valued Young integralshttps://zbmath.org/1527.260172024-02-28T19:32:02.718555Z"Michta, Mariusz"https://zbmath.org/authors/?q=ai:michta.mariusz"Motyl, Jerzy"https://zbmath.org/authors/?q=ai:motyl.jerzyLet \((X, \|\cdot\|)\) be a Banach space, \(\mathrm{Comp} (X)\) be the family of all nonempty and compact subsets of \(X\) and \(H\) be the Hausdorff metric in \(\mathrm{Comp}(X)\). For \(p \geq 1\), a set-valued function \(F: [0, T] \rightarrow \mathrm{Comp} (X)\) is said to have finite Riesz \(p\)-variation if there exists \(c > 0\) such that \(\sum_{i = 1}^n \frac{H^p\big(F(t_i) - F(t_{i-1}\big)}{(t_i - t_{i-1})^{p-1}} < c\) for all partitions \(0 = t_0 < t_1 < \dots < t_n = T\) of the interval \([0, T]\). For such multifunctions, set-valued integrals of Young type are introduced and selection results and properties of such set-valued integrals are discussed.
Reviewer: Mircea Balaj (Oradea)Fractional optimal reachability problems with \(\psi\)-Hilfer fractional derivativehttps://zbmath.org/1527.340242024-02-28T19:32:02.718555Z"Vellappandi, M."https://zbmath.org/authors/?q=ai:vellappandi.madasamy"Govindaraj, Venkatesan"https://zbmath.org/authors/?q=ai:govindaraj.venkatesan"Vanterler da C. Sousa, José"https://zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.jose(no abstract)Solution of a singular minimum energy control problem for time delay system: regularization approachhttps://zbmath.org/1527.341072024-02-28T19:32:02.718555Z"Glizer, Valery Y."https://zbmath.org/authors/?q=ai:glizer.valery-ySummary: A minimum energy control problem for a linear time-dependent differential system with state delays (point-wise and distributed) is considered. The feature of this problem is that a weight matrix of the control in the problem's functional is singular (but non-zero). Due to this feature, the problem itself is singular. Using the regularization method and the asymptotic analysis of the regularized minimum energy control problem, the solution to the considered singular minimum energy control problem is derived. Illustrative example is presented. Along with this example, an example on non-uniqueness of solution to the singular minimum energy control problem is presented.Approximate controllability of impulsive fractional evolution equations of order \(1<\alpha <2\) with state-dependent delay in Banach spaceshttps://zbmath.org/1527.341192024-02-28T19:32:02.718555Z"Arora, Sumit"https://zbmath.org/authors/?q=ai:arora.sumit"Mohan, Manil T."https://zbmath.org/authors/?q=ai:mohan.manil-t"Dabas, Jaydev"https://zbmath.org/authors/?q=ai:dabas.jaydev(no abstract)The distributional divergence of horizontal vector fields vanishing at infinity on Carnot groupshttps://zbmath.org/1527.350132024-02-28T19:32:02.718555Z"Baldi, A."https://zbmath.org/authors/?q=ai:baldi.annalisa"Montefalcone, F."https://zbmath.org/authors/?q=ai:montefalcone.francescopaoloSummary: We define a \(BV\)-type space in the setting of Carnot groups (i.e., simply connected Lie groups with stratified nilpotent Lie algebra) that allows one to characterize all distributions \(F\) for which there exists a continuous horizontal vector field \(\Phi\), vanishing at infinity, that solves the equation \(\operatorname{div}_H \Phi = F\). This generalizes to the setting of Carnot groups some results by \textit{T. De Pauw} and \textit{W. F. Pfeffer} [Commun. Pure Appl. Math. 61, No. 2, 230--260 (2008; Zbl 1137.35014)] and by \textit{T. De Pauw} and \textit{M. Torres} [Proc. R. Soc. Edinb., Sect. A, Math. 141, No. 1, 65--76 (2011; Zbl 1213.35181)] for the Euclidean setting.A perturbative approach to the parabolic optimal transport problemhttps://zbmath.org/1527.350282024-02-28T19:32:02.718555Z"Abedin, Farhan"https://zbmath.org/authors/?q=ai:abedin.farhan"Kitagawa, Jun"https://zbmath.org/authors/?q=ai:kitagawa.junSummary: Fix a pair of smooth source and target densities \(\rho\) and \(\rho^\ast\) of equal mass, supported on bounded domains \(\Omega, \Omega^\ast \subset \mathbb{R}^n\). Also fix a cost function \(c_0 \in C^{4,\alpha} (\overline{\Omega} \times \overline{\Omega^\ast})\) satisfying the weak regularity criterion of Ma, Trudinger, and Wang, and assume \(\Omega\) and \(\Omega^\ast\) are uniformly \(c_0\)- and \(c_0^\ast\)-convex with respect to each other. We consider a parabolic version of the optimal transport problem between \((\Omega,\rho)\) and \((\Omega^\ast,\rho^\ast)\) when the cost function \(c\) is a sufficiently small \(C^4\) perturbation of \(c_0\), and where the size of the perturbation depends on the given data. Our main result establishes global-in-time existence of a solution \(u \in C^2_xC^1_t(\overline{\Omega} \times [0, \infty))\) of this parabolic problem, and convergence of \(u(\cdot,t)\) as \(t \to \infty\) to a Kantorovich potential for the optimal transport map between \((\Omega,\rho)\) and \((\Omega^\ast,\rho^\ast)\) with cost function \(c\). This is the first convergence result for the parabolic optimal transport problem when the cost function \(c\) fails to satisfy the weak Ma-Trudinger-Wang condition by a quantifiable amount.Effective fronts of Polygon shapes in two dimensionshttps://zbmath.org/1527.350342024-02-28T19:32:02.718555Z"Jing, Wenjia"https://zbmath.org/authors/?q=ai:jing.wenjia"Tran, Hung V."https://zbmath.org/authors/?q=ai:tran.hung-vinh"Yu, Yifeng"https://zbmath.org/authors/?q=ai:yu.yifengSummary: We study the effective fronts of first order front propagations in two dimensions \((n=2)\) in the periodic setting. Using PDE-based approaches, we show that for every \(\alpha \in (0,1)\), the class of centrally symmetric polygons with rational vertices (i.e., vectors in \(\bigcup_{\lambda \in \mathbb{R}}\lambda \mathbb{Z}^2\)) and nonempty interior is admissible as effective fronts for front speeds in \(C^{1,\alpha}(\mathbb{T}^2,(0,\infty))\). This result can also be formulated in the language of stable norms corresponding to periodic metrics in \(\mathbb{T}^2\). Similar results were known long ago when \(n\geq 3\) for front speeds in \(C^{\infty} (\mathbb{T}^n,(0,\infty))\). The two-dimensional case is much more subtle due to topological restrictions. In fact, for given \(C^{1,1}(\mathbb{T}^2,(0,\infty))\) front speeds, the effective front is \(C^1\) and hence cannot be a polygon. Our regularity requirements on front speeds are hence optimal. To the best of our knowledge, this is the first time that polygonal effective fronts have been constructed in two dimensions.Symmetry breaking bifurcation of membranes with boundaryhttps://zbmath.org/1527.350392024-02-28T19:32:02.718555Z"Palmer, Bennett"https://zbmath.org/authors/?q=ai:palmer.bennett"Pámpano, Álvaro"https://zbmath.org/authors/?q=ai:pampano.alvaroSummary: We use a bifurcation theory due to Crandall and Rabinowitz to show the existence of a symmetry breaking bifurcation of a specific one parameter family of axially symmetric disc type solutions of a membrane equation with fixed boundary. In place of working directly with the fourth order membrane equation, it is replaced by a second order reduction found in [\textit{B. Palmer} and \textit{Á. Pámpano}, Calc. Var. Partial Differ. Equ. 61, No. 3, Paper No. 79, 28 p. (2022; Zbl 1490.49030)].Neural networks for first order HJB equations and application to front propagation with obstacle termshttps://zbmath.org/1527.351402024-02-28T19:32:02.718555Z"Bokanowski, Olivier"https://zbmath.org/authors/?q=ai:bokanowski.olivier.1|bokanowski.olivier.2|bokanowski.olivier"Prost, Averil"https://zbmath.org/authors/?q=ai:prost.averil"Warin, Xavier"https://zbmath.org/authors/?q=ai:warin.xavierSummary: We consider a deterministic optimal control problem, focusing on a finite horizon scenario. Our proposal involves employing deep neural network approximations to capture Bellman's dynamic programming principle. This also corresponds to solving first-order Hamilton-Jacobi-Bellman (HJB) equations. Our work builds upon the research conducted by \textit{C. Huré} et al. [SIAM J. Numer. Anal. 59, No. 1, 525--557 (2021; Zbl 1466.65007)], which primarily focused on stochastic contexts. However, our objective is to develop a completely novel approach specifically designed to address error propagation in the absence of diffusion in the dynamics of the system. Our analysis provides precise error estimates in terms of an average norm. Furthermore, we provide several academic numerical examples that pertain to front propagation models incorporating obstacle constraints, demonstrating the effectiveness of our approach for systems with moderate dimensions (e.g., ranging from 2 to 8) and for nonsmooth value functions.Optimal control of a parabolic equation with a nonlocal nonlinearityhttps://zbmath.org/1527.351622024-02-28T19:32:02.718555Z"Kenne, Cyrille"https://zbmath.org/authors/?q=ai:kenne.cyrille"Djomegne, Landry"https://zbmath.org/authors/?q=ai:djomegne.landry"Mophou, Gisèle"https://zbmath.org/authors/?q=ai:mophou.gisele-massengoSummary: This paper proposes an optimal control problem for a parabolic equation with a nonlocal nonlinearity. The system is described by a parabolic equation involving a nonlinear term that depends on the solution and its integral over the domain. We prove the existence and uniqueness of the solution to the system and the boundedness of the solution. Regularity results for the control-to-state operator, the cost functional and the adjoint state are also proved. We show the existence of optimal solutions and derive first-order necessary optimality conditions. In addition, second-order necessary and sufficient conditions for optimality are established.Existence of solutions to reaction cross diffusion systemshttps://zbmath.org/1527.351872024-02-28T19:32:02.718555Z"Jacobs, Matt"https://zbmath.org/authors/?q=ai:jacobs.matthewSummary: Reaction cross diffusion systems are a two species generalization of the porous media equation. These systems play an important role in the mechanical modeling of living tissues and tumor growth. Due to their mixed parabolic-hyperbolic structure, even proving the existence of solutions to these equations is challenging. In this paper, we exploit the parabolic structure of the system to prove the strong compactness of the pressure gradient in \(L^2\). The key ingredient is the energy dissipation relation, which, along with some compensated compactness arguments, allows us to upgrade weak convergence to strong convergence. As a consequence of the pressure compactness, we are able to prove the existence of solutions in a general setting and pass to the Hele-Shaw/incompressible limit in any dimension.Determination of rigid inclusions immersed in an isotropic elastic body from boundary measurementhttps://zbmath.org/1527.351892024-02-28T19:32:02.718555Z"Abdelwahed, Mohamed"https://zbmath.org/authors/?q=ai:abdelwahed.mohamed"Chorfi, Nejmeddine"https://zbmath.org/authors/?q=ai:chorfi.nejmeddine"Hassine, Maatoug"https://zbmath.org/authors/?q=ai:hassine.maatougSummary: We study the determination of some rigid inclusions immersed in an isotropic elastic medium from overdetermined boundary data. We propose an accurate approach based on the topological sensitivity technique and the reciprocity gap concept. We derive a higher-order asymptotic formula, connecting the known boundary data and the unknown inclusion parameters. The obtained formula is interesting and useful tool for developing accurate and robust numerical algorithms in geometric inverse problems.Systematic search for singularities in 3D Euler flowshttps://zbmath.org/1527.352512024-02-28T19:32:02.718555Z"Zhao, Xinyu"https://zbmath.org/authors/?q=ai:zhao.xinyu"Protas, Bartosz"https://zbmath.org/authors/?q=ai:protas.bartoszSummary: We consider the question whether starting from a smooth initial condition 3D inviscid Euler flows on a periodic domain may develop singularities in a finite time. Our point of departure is the well-known result by \textit{T. Kato} [J. Funct. Anal. 9, 296--305 (1972; Zbl 0229.76018)], which asserts the local existence of classical solutions to the Euler system in the Sobolev space \(H^m\) for \(m>5/2\). Thus, the potential formation of a singularity must be accompanied by an unbounded growth of the \(H^m\) norm of the velocity field as the singularity time is approached. We perform a systematic search for ``extreme'' Euler flows that may realize such a scenario by formulating and solving a PDE-constrained optimization problem where the \(H^3\) norm of the solution at a certain fixed time \(T > 0\) is maximized with respect to the initial data subject to suitable normalization constraints. This problem is solved using a state-of-the-art Riemannian conjugate gradient method where the gradient is obtained from solutions of an adjoint system. Computations performed with increasing numerical resolutions demonstrate that, as asserted by the theorem of Kato [loc. cit.], when the optimization time window \([0, T]\) is sufficiently short, the \(H^3\) norm remains bounded in the extreme flows found by solving the optimization problem, which indicates that the Euler system is well-posed on this ``short'' time interval. On the other hand, when the window \([0, T]\) is long, possibly longer than the time of the local existence asserted by Kato's theorem, then the \(H^3\) norm of the extreme flows diverges upon resolution refinement, which indicates a possible singularity formulation on this ``long'' time interval. The extreme flow obtained on the long time window has the form of two colliding vortex rings and is characterized by certain symmetries. In particular, the region of the flow in which a singularity might occur is nearly axisymmetric.Feedback control for fluid mixing via advectionhttps://zbmath.org/1527.352862024-02-28T19:32:02.718555Z"Hu, Weiwei"https://zbmath.org/authors/?q=ai:hu.weiwei"Rautenberg, Carlos N."https://zbmath.org/authors/?q=ai:rautenberg.carlos-n"Zheng, Xiaoming"https://zbmath.org/authors/?q=ai:zheng.xiaomingSummary: This work is concerned with nonlinear feedback control design for the problem of fluid mixing via advection. The overall dynamics is governed by the transport and Stokes equations in an open bounded and connected domain \(\Omega \subset \mathbb{R}^d\), with \(d = 2\) or \(d = 3\). The feedback laws are constructed based on the ideas of instantaneous control as well as a direct approximation of the optimality system derived from an optimal open-loop control problem. It can be shown that under appropriate numerical discretization schemes, two approaches generate the same sub-optimal feedback law. On the other hand, different discretization schemes may result in feedback laws of different regularity, which determine different mixing results. The Sobolev norm of the dual space \(( H^1 ( \Omega ) )^\prime\) of \(H^1(\Omega)\) is used as the mix-norm to quantify mixing based on the known property of weak convergence. The major challenge is encountered in the analysis of the asymptotic behavior of the closed-loop systems due to the absence of diffusion in the transport equation together with its nonlinear coupling with the flow equations. To address these issues, we first establish the decay properties of the velocity, which in turn help obtain the estimates on scalar mixing and its long-time behavior. Finally, mixed continuous Galerkin (CG) and discontinuous Galerkin (DG) methods are employed to discretize the closed-loop system. Numerical experiments are conducted to demonstrate our ideas and compare the effectiveness of different feedback laws.BV entropy solutions of two-dimensional nonstationary Prandtl boundary layer systemhttps://zbmath.org/1527.353112024-02-28T19:32:02.718555Z"Zhan, Huashui"https://zbmath.org/authors/?q=ai:zhan.huashuiSummary: A new kind of BV entropy solution matching up with the degenerate parabolic equation arising from the two-dimensional Prandtl boundary layer system is introduced. If there are some restrictions on the coefficients of the system, then by means of Kruzkov's bi-variables method, the stability of entropy solutions is proved independent of the boundary value condition. For some domains which are regular in a special sense, by the inverse transformation of the Crocco transformation, the two-dimensional Prandtl boundary layer systems are well-posedness.Normalized solutions for the Klein-Gordon-Dirac systemhttps://zbmath.org/1527.353242024-02-28T19:32:02.718555Z"Zelati, Vittorio Coti"https://zbmath.org/authors/?q=ai:coti-zelati.vittorio"Nolasco, Margherita"https://zbmath.org/authors/?q=ai:nolasco.margheritaSummary: We prove the existence of a stationary solution for the system describing the interaction between an electron coupled with a massless scalar field (a photon). We find a solution, with fixed \(L^2\)-norm, by variational methods, as a critical point of an energy functional.Existence and instability of standing wave for the two-wave model with quadratic interactionhttps://zbmath.org/1527.353772024-02-28T19:32:02.718555Z"Gan, Zaihui"https://zbmath.org/authors/?q=ai:gan.zaihui"Wang, Yue"https://zbmath.org/authors/?q=ai:wang.yue.11|wang.yue.4|wang.yue.2|wang.yue.16|wang.yue.1|wang.yue.3|wang.yue.5Summary: In this paper, we establish the existence and instability of standing wave for a system of nonlinear Schrödinger equations arising in the two-wave model with quadratic asymmetric interaction in higher space dimensions under mass resonance conditions. Here, we relax the limitation for the relationship between complex constants \(a_1\) and \(a_2\) given in [\textit{N. Hayashi} et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 30, No. 4, 661--690 (2013; Zbl 1291.35347)], and consider arbitrary real positive constants \(a_1\) and \(a_2\). First of all, according to the conservation quantities of mass and energy, using the so-called virial type estimate, we obtain that the solution of the Cauchy problem under consideration blows up in finite time in \(H^1(\mathbb{R}^N)\times H^1(\mathbb{R}^N)\) with space dimension \(N \geq 4\). Next, for space dimension \(N\) with \(4 < N < 6\), we establish the existence of the ground state solution for the elliptic equations corresponding to the nonlinear Schrödinger equations under frequency and mass resonances by adopting variational methods, and further achieve the exponential decay at infinity for the ground state. This implies the existence of standing wave for the nonlinear Schrödinger equaitons under consideration. Finally, by defining another constrained minimizing problem for a pair of complex-valued functions, a suitable manifold, referring to the characterization of the standing wave, making appropriate scaling and adopting virial type estimate, we attain the instability of the standing wave for the equations under frequency and mass resonances in space dimension \(N\) with \(4 < N < 6\) by virtue of the conservations of mass and energy. Here, we adopt the equivalence of two constrained minimizing problems defined for pairs of complex-valued and real-valued functions \((u, v)\), respectively, when \((u, v)\) is a pair of real-valued functions.Large deviations principle for the cubic NLS equationhttps://zbmath.org/1527.353782024-02-28T19:32:02.718555Z"Garrido, Miguel Angel"https://zbmath.org/authors/?q=ai:garrido.miguel-angel"Grande, Ricardo"https://zbmath.org/authors/?q=ai:grande.ricardo"Kurianski, Kristin M."https://zbmath.org/authors/?q=ai:kurianski.kristin-m"Staffilani, Gigliola"https://zbmath.org/authors/?q=ai:staffilani.gigliolaSummary: In this paper, we present a probabilistic study of rare phenomena of the cubic nonlinear Schrödinger equation on the torus in a weakly nonlinear setting. This equation has been used as a model to numerically study the formation of rogue waves in deep sea. Our results are twofold: first, we introduce a notion of criticality and prove a Large Deviations Principle (LDP) for the subcritical and critical cases. Second, we study the most likely initial conditions that lead to the formation of a rogue wave, from a theoretical and numerical point of view. Finally, we propose several open questions for future research.
{\copyright} 2023 Wiley Periodicals LLC.Positive ground state solutions for generalized quasilinear Schrödinger equations with critical growthhttps://zbmath.org/1527.353902024-02-28T19:32:02.718555Z"Meng, Xin"https://zbmath.org/authors/?q=ai:meng.xin"Ji, Shuguan"https://zbmath.org/authors/?q=ai:ji.shuguanSummary: This paper concerns the existence of positive ground state solutions for generalized quasilinear Schrödinger equations in \(\mathbb{R}^N\) with critical growth which arise from plasma physics, as well as high-power ultrashort laser in matter. By applying a variable replacement, the quasilinear problem reduces to a semilinear problem which the associated functional is well defined in the Sobolev space \(H^1 (\mathbb{R}^N)\). We use the method of Nehari manifold for the modified equation, establish the minimax characterization, and then prove that each Palais-Smale sequence of the associated energy functional is bounded. By combining Lions's concentration-compactness lemma together with some classical arguments developed by \textit{H. Brézis} and \textit{L. Nirenberg} [Commun. Pure Appl. Math. 36, 437--477 (1983; Zbl 0541.35029)], we obtain that the bounded Palais-Smale sequence has a nonvanishing behavior. Finally, we establish the existence of a positive ground state solution under some appropriate assumptions.Birth-death dynamics for sampling: global convergence, approximations and their asymptoticshttps://zbmath.org/1527.354052024-02-28T19:32:02.718555Z"Lu, Yulong"https://zbmath.org/authors/?q=ai:lu.yulong"Slepčev, Dejan"https://zbmath.org/authors/?q=ai:slepcev.dejan"Wang, Lihan"https://zbmath.org/authors/?q=ai:wang.lihanSummary: Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we study a continuum birth-death dynamics. We improve results in previous works
[\textit{L. Liu} et al., Appl. Math. Optim. 87, No. 3, Paper No. 48, 27 p. (2023; Zbl 1515.49025); \textit{Y. Lu}, \textit{J. Lu} and \textit{J. Nolen}, ``Accelerating Langevin sampling with birth-death'', Preprint, \url{arXiv:1905.09863}]
and provide weaker hypotheses under which the probability density of the birth-death governed by Kullback-Leibler divergence or by \(\chi^2\) divergence converge exponentially fast to the Gibbs equilibrium measure, with a universal rate that is independent of the potential barrier. To build a practical numerical sampler based on the pure birth-death dynamics, we consider an interacting particle system, which is inspired by the gradient flow structure and the classical Fokker-Planck equation and relies on kernel-based approximations of the measure. Using the technique of \(\Gamma\)-convergence of gradient flows, we show that on the torus, smooth and bounded positive solutions of the kernelised dynamics converge on finite time intervals, to the pure birth-death dynamics as the kernel bandwidth shrinks to zero. Moreover we provide quantitative estimates on the bias of minimisers of the energy corresponding to the kernelised dynamics. Finally we prove the long-time asymptotic results on the convergence of the asymptotic states of the kernelised dynamics towards the Gibbs measure.
{{\copyright} 2023 IOP Publishing Ltd \& London Mathematical Society}Synchronization in a Kuramoto mean field gamehttps://zbmath.org/1527.354272024-02-28T19:32:02.718555Z"Carmona, Rene"https://zbmath.org/authors/?q=ai:carmona.rene-a"Cormier, Quentin"https://zbmath.org/authors/?q=ai:cormier.quentin"Soner, H. Mete"https://zbmath.org/authors/?q=ai:soner.halil-meteSummary: The classical Kuramoto model is studied in the setting of an infinite horizon mean field game. The system is shown to exhibit both synchronization and phase transition. Incoherence below a critical value of the interaction parameter is demonstrated by the stability of the uniform distribution. Above this value, the game bifurcates and develops self-organizing time homogeneous Nash equilibria. As interactions get stronger, these stationary solutions become fully synchronized. Results are proved by an amalgam of techniques from nonlinear partial differential equations, viscosity solutions, stochastic optimal control and stochastic processes.On numerical approximations of fractional and nonlocal mean field gameshttps://zbmath.org/1527.354282024-02-28T19:32:02.718555Z"Chowdhury, Indranil"https://zbmath.org/authors/?q=ai:chowdhury.indranil"Ersland, Olav"https://zbmath.org/authors/?q=ai:ersland.olav"Jakobsen, Espen R."https://zbmath.org/authors/?q=ai:jakobsen.espen-robstadSummary: We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove convergence along subsequences for (i) degenerate equations in one space dimension and (ii) nondegenerate equations in arbitrary dimensions. We also give results on full convergence and convergence to classical solutions. Numerical tests are implemented for a range of different nonlocal diffusions and support our analytical findings.Optimal control of fractional Sturm-Liouville wave equations on a star graphhttps://zbmath.org/1527.354572024-02-28T19:32:02.718555Z"Moutamal, Maryse M."https://zbmath.org/authors/?q=ai:moutamal.maryse-m"Joseph, Claire"https://zbmath.org/authors/?q=ai:joseph.claireSummary: In the present paper, we are concerned with a fractional wave equation of Sturm-Liouville type in a general star graph. We first give several existence, uniqueness and regularity results of weak solutions for the one-dimensional case using the spectral theory; we prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal control via the Euler-Lagrange first-order optimality conditions. We then investigate the analogous problems for a fractional Sturm-Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary conditions and controls of the velocity. We show the existence and uniqueness of minimizers, and by using the first-order optimality conditions with the Lagrange multipliers, we are able to characterize the optimal controls.On a class of nonlocal problems with fractional gradient constrainthttps://zbmath.org/1527.354632024-02-28T19:32:02.718555Z"Azevedo, Assis"https://zbmath.org/authors/?q=ai:azevedo.assis"Rodrigues, José-Francisco"https://zbmath.org/authors/?q=ai:rodrigues.jose-francisco"Santos, Lisa"https://zbmath.org/authors/?q=ai:santos.lisaSummary: We consider a Hilbertian and a charges approach to fractional gradient constraint problems of the type \(|D^\sigma u|\leq g\), involving the distributional fractional Riesz gradient \(D^\sigma\), \(0 < \sigma < 1\), extending previous results on the existence of solutions and Lagrange multipliers of these nonlocal problems.
We also prove their convergence as \(\sigma\nearrow 1\) towards their local counterparts with the gradient constraint \(|D u|\leq g\).
For the entire collection see [Zbl 1519.00033].Analysis of nonlinear systems arise in thermoelasticity using fractional natural decomposition schemehttps://zbmath.org/1527.354792024-02-28T19:32:02.718555Z"Sarwe, Deepak Umrao"https://zbmath.org/authors/?q=ai:sarwe.deepak-umrao"Kulkarni, Vinayak S."https://zbmath.org/authors/?q=ai:kulkarni.v-s(no abstract)On the topological gradient method for an inverse problem resolutionhttps://zbmath.org/1527.354872024-02-28T19:32:02.718555Z"Abdelwahed, Mohamed"https://zbmath.org/authors/?q=ai:abdelwahed.mohamed"Chorfi, Nejmeddine"https://zbmath.org/authors/?q=ai:chorfi.nejmeddineSummary: In this work, we consider the topological gradient method to deal with an inverse problem associated with three-dimensional Stokes equations. The problem consists in detecting unknown point forces acting on fluid from measurements on the boundary of the domain. We present an asymptotic expansion of the considered cost function using the topological sensitivity analysis method. A detection algorithm is then presented using the developed formula. Some numerical tests are presented to show the efficiency of the presented algorithm.Homogenization for sub-Riemannian Lagrangianshttps://zbmath.org/1527.370682024-02-28T19:32:02.718555Z"Sánchez Morgado, Héctor"https://zbmath.org/authors/?q=ai:sanchez-morgado.hectorThe author studies the problem of homogenization of a class of Hamilton-Jacobi equations. The usual tool in this framework is weak KAM theory. In this paper the author proves a version of the Tonelli theorem in the sub-Riemannian context, thus enabling him to use several known geometric results. The main achievements concern homogenization by means of the Lax-Oleinik formula, which relates to viscosity solutions of the considered initial value problem. The paper is well written and organized, but there are a few typos in the text.
Reviewer: Ram Verma (Balrampur)Invariant ideals and their applications to the turnpike theoryhttps://zbmath.org/1527.370992024-02-28T19:32:02.718555Z"Mammadov, Musa"https://zbmath.org/authors/?q=ai:mammadov.musa-a"Szuca, Piotr"https://zbmath.org/authors/?q=ai:szuca.piotrSummary: In this paper, the turnpike property is established for a nonconvex optimal control problem in discrete time. The functional is defined by the notion of the ideal convergence and can be considered as an analogue of the terminal functional defined over infinite-time horizon. The turnpike property states that every optimal solution converges to some unique optimal stationary point in the sense of ideal convergence if the ideal is invariant under translations. This kind of convergence generalizes, for example, statistical convergence and convergence with respect to logarithmic density zero sets.An invitation to optimal transport. Wasserstein distances, and gradient flowshttps://zbmath.org/1527.490012024-02-28T19:32:02.718555Z"Figalli, Alessio"https://zbmath.org/authors/?q=ai:figalli.alessio"Glaudo, Federico"https://zbmath.org/authors/?q=ai:glaudo.federicoThis is the second edition of a graduate text which gives an introduction to optimal transport theory. The first chapter gives an introduction of the historical roots of optimal transport, with the work of Gaspard Monge and Leonid Kantorovich. Moreover the basic notions of measure theory and Riemannian Geometry are presented. Finally some examples of transport maps are presented.
Chapter 2 presents the core of optimal transport theory, as the solution of Kantorovich's problem for general costs and the solution of the Monge's problem for suitable costs. Other applications are presented, as the polar decomposition and an application to the Euler equation of fluid dynamics.
Chapter 3 presents some connections between optimal transport, gradient flows and partial differential equations. The Wasserstein distances and gradient flows in Hilbert spaces are introduced. Then the authors show that the gradient flow of the entropy functional in the Wasserstein space coincides with the heat equation, following the seminal approach of Jordan, Kinderlehrer and Otto.
Chapter 4 is devoted to an analysis of optimal transport from the differential point of view, in particular some several important partial differential equations are interpreted as gradient flows with respect to the 2-Wasserstein distance.
The last Chapter 5 presents some further reading on optimal transport for the readers.
The book contains also two appendices, Appendix A, which presents some exercises on optimal transport, and Appendix B, in which the authors give a sketch of the proof of a disintegration theorem, remanding to a book by Luigi Ambrosio, Nicola Fusco e Diego Pallara for a complete proof.
For the first edition of the book, see [\textit{A. Figalli} and \textit{F. Glaudo}, An invitation to optimal transport, Wasserstein distances, and gradient flows. Berlin: European Mathematical Society (EMS) (2021; Zbl 1472.49001)].
Reviewer: Antonio Masiello (Bari)Optimization under stochastic uncertainty. Methods, control and random search methodshttps://zbmath.org/1527.490022024-02-28T19:32:02.718555Z"Marti, Kurt"https://zbmath.org/authors/?q=ai:marti.kurtThe book is dedicated to optimization problems under stochastic uncertainty and contains four parts, 18 chapters and 3 appendices.
Part 1 describes stochastic optimization tools. It is divided in 5 chapters.
In Chapter 1, the author starts introducing the basic control system of first-order differential equations with random parameters written as: \( \overset{.}{z}(t)=g(t,\omega ,z(t),u(t))\), for \(t_{0}\leq t\leq t_{f}\), and \( \omega \in \Omega \). Here, \(z=z(t,\omega )\in \mathbb{R}^{m}\) is the state variable, \(\omega \) takes its values in a probability space \((\Omega , \mathcal{A},P)\), \(\Omega \)\ being the set of elementary events, \(\mathcal{A}\) the \(\sigma \)-algebra of events and \(P\) the probability measure, \(u(t)\in \mathbb{R}^{n}\) is the deterministic or stochastic control input and belongs to a suitable linear space \(U\), and the function \(g:[t_{0},t_{f}]\times \Omega \times \mathbb{R}^{m}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{m}\) is at least continuous. The initial condition \(z(t_{0})=z_{0}(\omega )\in \mathbb{R}^{m}\) is imposed, where \(z_{0}\) is the random initial state. The author observes that this problem may be represented as an integral equation. He introduces an \(r\)-dimensional stochastic process \(\theta =\theta (t,\omega )\), takes \(g(t,\omega ,z,u)=\widetilde{g}(t,\theta (t,\omega ),z,u) \), where \(\widetilde{g}:[t_{0},t_{f}]\times \mathbb{R}^{r}\times \mathbb{R}^{m}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{m}\) is a continuous function having continuous Jacobians \(D_{\theta }\widetilde{g}\), \( D_{z}\widetilde{g}\), and \(D_{u}\widetilde{g}\) with respect to \(\theta ,z,u\), \(U\) as the Banach space of all regulated functions \(u(\cdot ):[t_{0},t_{f}]\rightarrow \mathbb{R}^{n}\), normed by the supremum norm \( \left\Vert \cdot \right\Vert _{\infty }\), \(Z=C_{0}^{m}[t_{0},t_{f}]\) and \( \theta (\cdot ,\omega )\in \Theta =C_{0}^{r}[t_{0},t_{f}]\) a.s., and he considers the problem: \(\overset{.}{z}(t)=\widetilde{g}(t,\overline{\theta } (t),z(t),\overline{u}(t))\), for \(t_{0}\leq t\leq t_{f}\), with the initial condition \(z(t_{0})=\overline{z}_{0}(\omega )\in \mathbb{R}^{m}\). Under regularity and boundedness properties, this initial value problem has a unique solution. He introduces the objective function \(F=F(u(\cdot ))=Ef(\omega ,S(\omega ,u(\cdot )),u(\cdot ))\), where \(E=E(\cdot \mid \mathcal{A}_{t_{0}})\) denotes the conditional expectation given the information \(\mathcal{A}_{t_{0}}\) about the control process up to the considered starting time point \(t_{0}\), and \(f=f(\omega ,z(\cdot ),u(\cdot ))=\int_{t_{0}}^{t_{f}}L(t,\omega ,z(t),u(t))dt+G(t_{f},\omega ,z(t_{f}))\) denotes the stochastic total costs arising along the trajectory \( z=z(t,\omega )\) and at the terminal point \(z_{f}=z(t_{f},\omega )\), \(z(\cdot )\in Z\), \(u(\cdot )\in U\). Here \(L:[t_{0},t_{f}]\times \Omega \times \mathbb{ R}^{m}\times \mathbb{R}^{n}\rightarrow \mathbb{R}\) and \(G:[t_{0},t_{f}] \times \Omega \times \mathbb{R}^{m}\rightarrow \mathbb{R}\) are given measurable cost functions. The author assumes that \(L(t,\omega ,\cdot ,\cdot )\) and \(G(t,\omega ,\cdot )\) are convex functions for each \((t,\omega )\in \lbrack t_{0},t_{f}]\times \Omega \), having continuous partial derivatives \( \nabla _{z}L(\cdot ,\omega ,\cdot ,\cdot )\), \(\nabla _{u}L(\cdot ,\omega ,\cdot ,\cdot )\) and \(\nabla _{z}G(\cdot ,\omega ,\cdot )\). The optimal control under stochastic uncertainty consists to find optimal controls being robust with respect to stochastic parameter variations \(u^{\ast }(\cdot )\), \( u^{\ast }(\cdot ,\cdot )\), satisfying \(\min F(u(\cdot ))\) such that \(u(\cdot )\in D\), the set of admissible controls. The author introduces an equivalent problem assuming further hypotheses on the data. He then presents three control laws: an open-loop control, assuming that the control \(u\) is deterministic, a closed-loop control or feedback control with a stochastic function \(u\), an open-loop feedback, or control/stochastic open-loop feedback, and a nonlinear model predictive control/stochastic nonlinear model. He shows how to use Taylor expansions to derive approximate control problems, that he illustrates in the case of a linear or sublinear cost function and finally on the above-indicated examples.
Chapter 2 is devoted to the study of regulator in the present context of stochastic optimization. The Chapter starts with an example of an active control under stochastic uncertainty. The author introduces the dynamic equation of the underlying control system \(F(p_{D},q(t),\overset{.}{q}(t), \overset{..}{q}(t))=u(t)\), \(t\geq t_{0}\), with the initial conditions \( q(t_{0})=q_{0}\), \(\overset{.}{q}(t_{0})=\overset{.}{q}_{0}\), where \(q=q(t)\) denotes the vector of configuration variables and \(u=u(t)\) is the control or input vector. The state trajectory of the system \(z(t)=\left( \begin{array}{c} q(t) \\
\overset{.}{q}(t) \end{array} \right) \), \(t_{0}\leq t\leq t_{f}\), is the solution to the preceding equation and initial conditions, which is related to the initial state \( z_{0}=\left( \begin{array}{c} q_{0} \\
\overset{.}{q}_{0} \end{array} \right) \), the control or input vector \(u=u(t)\), and the vector of dynamic parameters \(p_{D}\). The control function \(u=u(t)\), \(t_{0}\leq t\leq t_{f}\), is represented by: \(u(t)=u^{R}(t)+\Delta u(t)\), \(t_{0}\leq t\leq t_{f}\), where \(u^{R}=u^{R}(t)\), \(t_{0}\leq t\leq t_{f}\), denotes the feedforward control and \(u=u(t)\), \(t_{0}\leq t\leq t_{f}\), is a control correction, represented by a feedback control. The PID-regulator is defined as: \(\Delta u(t)=\varphi (t,\Delta q(t),\Delta q_{I}(t),\Delta \overset{.}{q}(t))\), \( t_{0}\leq t\leq t_{f}\), where \(q_{I}(t)=\int_{t_{0}}^{t}q(\tau )d\tau \), \( t_{0}\leq t\leq t_{f}\), denotes the integrated position and \(\Delta q_{I}(t)=\int_{t_{0}}^{t}\Delta q(\tau )d\tau \) is the deviation of the integrated position. The author draws computations on the errors, that he specializes in the case of quadratic cost functions, especially using Taylor expansions. This finally leads to an approximate regulator optimization problem, and the author ends the Chapter with an example.
In Chapter 3, the author focuses on optimal open-loop control problems of dynamic systems under stochastic uncertainty. He defines a stochastic optimal open-loop control \(u^{\ast }=u^{\ast }(t;t_{0},z_{0})\), \(t_{0}\leq t\leq t_{f}\), as a solution to the stochastic optimization problem:
\[
\min E\left( \int_{t_{0}}^{t_{f}}L(t,a(\omega ),q(t),\overset{.}{q} (t),u(t,z_{0}(\omega ))dt+G(a(\omega ),q(t_{f})\mid \mathcal{A} _{t_{0}}\right),
\]
such that: \(F(t,p(\omega ),q(t),\overset{.}{q}(t),\overset {..}{q}(t),u(t,z_{0}(\omega )))=0\), \(t_{0}\leq t\leq t_{f}\), a.s., \( q(t_{0},\omega )=q_{0}(\omega )\), \(\overset{.}{q}(t_{0},\omega )=\overset{.}{ q}_{0}(\omega )\), \(u(t)\in U_{t}\), \(t_{0}\leq t\leq t_{f}\), where the open-loop control \(u(t)=u(t;t_{0},z_{0})\) depends on the initial time \(t_{0}\) and on the initial state \(z_{0}=\left( \begin{array}{c} q_{0} \\
\overset{.}{q}_{0} \end{array} \right) \). Under a special choice of the cost functions \(L\) and \(G\), the author builds approximations of the expectation of the cost functions, which simplify the optimization problem.
In the short Chapter 4, the author uses homotopy methods for the construction of feedback controls. He writes the dynamic equation of the control system as the first-order initial value problem: \(\overset{.}{z} (t)=f(t,\theta (\omega ),z(t),u(t))\), \(t_{0}\leq t\leq t_{f}\), \( z(t_{0})=z_{0}(\omega )\). Considering the construction of a state-feedback control \(u(t)=\varphi (t,z(t))\), \(t\geq t_{0}\), where \(\varphi =\varphi (t,z) \) is a feedback control law, the construction of the homotopy theory is based on a transfer \(u(t)=\varphi _{\epsilon }(t,z)=\varphi _{0}(t,z)+\epsilon (\varphi (t,z)-\varphi _{0}(t,z))\), \(0\leq \epsilon \leq 1 \), from an open-loop control (\(\epsilon =0\)) \(u_{OL}=\varphi _{0}(t,z)=u_{0}(t,z_{0})\), \(t_{0}\leq t\leq t_{f}\),.with a time-function \( u_{0}=u_{0}(t,z_{0})\), to a feedback control (\(\epsilon =1\)) \(u_{FB}=\varphi =\varphi (t,z)\), \(t_{0}\leq t\leq t_{f}\). This leads to the initial value problem: \(\overset{.}{z}(t)=f(t,\theta (\omega ),z(t),\varphi (t,z))\), \( t_{0}\leq t\leq t_{f}\), \(z(t_{0})=z_{0}(\omega )\). After some computations, the author obtains the optimal feedback control problem under stochastic uncertainty minimizing the cost \(\min E\left( \int_{t_{0}}^{t_{f}}L(t,\theta (\omega ),z(t,a(\omega )),\varphi (t,z(t,a(\omega ))))dt+G(\theta (\omega ),z(t_{f}a(\omega )))\mid \mathcal{A}_{t_{0}}\right) \), subject to the preceding first-order initial value problem and with the feedback control \(u\) . He decomposes the optimization of the feedback control into two stages.
The long Chapter 5 considers a mathematical model of a technical or economic system or structure that is based on a state vector \(y\) containing the minimum number of internal state variables needed to describe the properties of the device.\ A vector \(a\) of model parameters, a vector \(x\) of nominal design variables and, in some cases, a vector \(u\) of control variables are also introduced. The vector of model parameters \(a=a(\omega )\) is represented by a random vector on a probability space \((\Omega ,\mathcal{A} ,P)\). In the reliability theory of structural systems, the limit state function \(g=g(a,x)\) has been introduced to separate, for a given design vector \(x\), the safe and unsafe domains. An initial representation of the limit state function \(g\) is: \(g(a)=\gamma (R(a)-L(a))\), where \(R=R(a,\cdot )\) is the structural resistance, \(L=L(a,\cdot )\) the external loading, and \( \gamma \) an appropriate function. The chapter presents an optimization-oriented approach for the construction of an appropriate limit state functions. The author introduces the state equation \(T(y,u,(a(\omega ),x))=0\), \(y\in Y_{0}\subset \mathbb{R}^{m_{y}}\), \(u\in U_{0}\), where \( T=T(y,u,(a,x))\) is a linear or nonlinear mapping from \(\mathbb{R} ^{m_{y}}\times \mathbb{R}^{n}\) into \(\mathbb{R}^{m_{T}}\), and the admissibility condition \(y\in Y(u,a(\omega ),x)\) or \(y\in \mathrm{int}(Y(u,a(\omega ),x))\), where \(Y(u,a(\omega ),x)\) is the admissible state domain, defined as \(Y(u,a(\omega ),x)=\{y\in \mathbb{R}^{m_{y}}:g(u,a(\omega ),x)\leq 0\}\), \(g\) \ being a scalar or \(m_{g}\)-vector response function having appropriate analytical properties (e.g., convexity, linearity) and being usually affine-linear with respect to \(y\). Assuming that the feasible state domain \( Y(u,a(\omega ),x)\) is a closed and convex set containing the zero state as an interior point, the authors recalls that the admissibility condition can be represented by \(\pi (y\mid Y(u,a(\omega ),x))\leq 1\), or \(\pi (y\mid Y(u,a(\omega ),x))<1\), where \(\pi (y\mid Y(u,a(\omega ),x))\) denotes the Minkowski distance functional to the admissible state domain \(Y(u,a(\omega ),x)\) defined by \(\pi (y\mid Y)=inf\{\lambda >0:\frac{y}{\lambda }\in Y\}\). The author writes four optimization problems whose minimum value function is the state function \(s^{\ast }=s^{\ast }(a(\omega ),x)\). He defines the notion of safe states and he establishes characterizations of such safe states. He gives examples of systems with safe states, including situations which involve the time parameter.
Part 2 is mainly devoted to random search methods. It is divided in 3 chapters, starting from Chapter 6.
In Chapter 6, the author considers global optimization problems \(\min F(x)\) such that \(x\in D\), where \(D\) is a subset of \(\mathbb{R}^{n}\).\ He recalls the basic random search routine constructed according to the recurrence relation \(X_{t+1}=z_{t+1}\), if \(z_{t+1}\in D\) and \(F(z_{t+1})<F(X_{t})\) or \( X_{t+1}=X_{t}\), if \(z_{t+1}\notin D\) or \(F(z_{t+1})\geq F(X_{t})\), \( t=0,1,2,\ldots \), from a starting point \(X_{0}\). He proves convergence properties of this basic random search routine in terms of the set \( B_{\varepsilon ,M}=\{y\in D:F(y)\leq F^{\ast }+\varepsilon \), if \(F^{\ast }\in \mathbb{R}\), and \(F(y)\leq M\), if \(F^{\ast }=-\infty \}\), where \( F^{\ast }=inf\{F(x):x\in D\}\). He then considers discrete optimization problems, assuming that \(D\) contains a finite number of elements in \(\mathbb{ R}^{n}\) and he proves a convergence result. He proposes adaptive random search methods, introducing a set of admissible decision rules. The Chapter ends with a deeper analysis of the minimization of a real valued convex function \(F:\mathbb{R}\rightarrow \mathbb{R}\) with respect to \(D=\mathbb{R}\).
Chapter 7 describes controlled random search methods under uncertainty, which increase the rate of convergence of the basic search method. They are based on transition probabilities \(\pi _{t}(x^{t},\cdot )=\pi _{t}(a,x^{t},\cdot )\) which depend on parameters \(a=(a_{j})_{j\in J}\in A\), to be chosen in an optimal way. A conditional mean gain is computed at each step, that involves the area of success \(G_{F}(x_{t})\) which can approximated using a random search Newton method though an approximation of the value \(F(z_{t+1})\) by a second-order Taylor polynomial, or a sequential stochastic decision process under uncertainty and a Bayesian model for the unknown \(F\). The author proves a convergence result for the proposed controlled random search method.
Chapter 8 presents controlled random search procedures for global optimization. The author introduces the random search method that builds the sequence \(X_{0}(\omega ),X_{1}(\omega ),\ldots ,X_{n}(\omega ),\ldots \) of random iterates according to \(X_{n+1}(\omega )=z_{n+1}\), if \(z_{n+1}\in D\) and \(F(z_{n+1})<F(X_{n}(\omega ))\) and \(X_{n+1}(\omega )=X_{n}(\omega )\), otherwise, where \(x_{0}(\omega )=x_{0}\in D\) is a given starting point in \(D\) and \(z_{1},z_{2},\ldots ,z_{n},\ldots \) are realizations of a sequence of random \(d\)-vectors with conditional distributions involving transition probability measures to be selected. He proves a convergence result under appropriate hypotheses. He then describes controlled random search methods that associate with the random search routine described in Chapter 8 a sequential stochastic decision process. He draws computations on the optimal controls and he establishes convergence rates of such controlled random search procedures. The Chapter ends with the description of numerical realizations of optimal control laws, through a modified quasi-Newton condition.
Part 3 analyzes the convergence and convergence rates of random search methods. It is divided in 9 chapters, from Chapter 9 to Chapter 17.
Chapter 9 starts with the problem: \(\min_{x\in D}f(x)\), where \(f:D\rightarrow \mathbb{R}\) and \(D\subset \mathbb{R}^{d}\) not empty. The author assumes that \(D\) and \(f\) are measurable. He writes the search problem as \(x_{n}=y_{n}\) if \(y_{n}\in D\) and \(f(y_{n})<f(x_{n-1})\), \(x_{n}=x_{n-1}\) otherwise, or as \( x_{n}=y_{n}\cdot 1_{\{x\in D:f(x)<f(x_{n-1})\}}(y_{n})+x_{n-1}\cdot 1_{ \mathbb{R}^{d}\setminus \{x\in D:f(x)<f(x_{n-1})\}}(y_{n})\), \(n\in \mathbb{N} \). He generalizes this presentation introducing a sequence of transition probabilities \(p_{n}\) from \((\mathbb{R}^{d})^{n}\) to \(\mathbb{R}^{d}\), called mutation sequence. He associates a selection sequence and he establishes properties of these sequences.
Chapter 10 presents random search methods involving an absolutely continuous mutation sequence, or random direction methods, and the links between these methods.
Chapter 11 proves accessibility results concerning the convergence of the random search procedure: \(f(X_{n})\rightarrow \inf_{x\in D}f(x)=f^{\ast }\), \(P \)- a.s., \(n\rightarrow \infty \), for all starting points \(x_{0}\in D\). The author defines \(D^{\ast }=\{x\in D:f(x)=f^{\ast }\}\) and \(D_{a}=\{x\in D:f(x)\leq a\}\), for every \(a\in \mathbb{R}\). \((X_{n}:n\in \mathbb{N}_{0})\) being a random search method with an arbitrary mutation sequence \( (p_{n}:n\in \mathbb{N})\) and an arbitrary starting point \(x_{0}\), he proves equivalent formulations of the convergence \(\lim_{n\rightarrow \infty }P(X_{n}\in D_{a})=1\), and sufficient conditions for this limit to occur.
Chapter 12 proves convergence results for the random search methods presented in Chapter 10. The results are illustrated with examples.
Chapter 13 proves convergence results for stationary random search methods for positive success probability.
Chapter 14 presents random search methods with convergence order \( O(n^{-\alpha })\) and Chapter 15 presents random search methods with a linear rate of convergence. Examples are studied which illustrate the convergence results.
Chapter 16 describes a random direction procedure based on a very simple step length control. The author proves convergence results.
Chapter 17 presents hybrid random search methods.
Part 4 is devoted to the resolution of optimization problems under stochastic uncertainty by random search methods and contains a unique chapter.
The author here considers optimization problems of the type: \(min_{x\in D}F_{0}(x)\), assuming that only statistical information is available about the objective function \(F_{0}\). Only a realization of a random function \( f=f(\omega ,x)\) on a probability space \((\Omega ,\mathcal{A},P)\) is available such that \(F_{0}(x)\) is the expectation of \(f(\cdot ,x):F_{0}(x)=Ef(\omega ,x)\), \(x\in D\). A special case is given by \(f(\omega ,x)=F_{0}(x)+n(\omega ,x)\), where \(n(\omega ,x)\) is an additional zero-mean random noise term. Given \(m\) independent sample functions \(f_{k}(x)=f(\omega _{k},x)\), \(k\in M\), of the random function \(F_{0}\), the estimated objective function \(\widehat{F}(x)=\frac{1}{m}\sum_{k\in M}f_{k}(x)\) is used and the basic random search procedure already considered in Part II and Part III is replaced by \(X_{t+1}=Z_{t+1}\), if \(Z_{t+1}\in D\) and \(\widehat{F} _{t}(Z_{t+1})<\widehat{F}_{t}(X_{t})\) and \(X_{t+1}=X_{t}\), if \(Z_{t+1}\notin D\) or \(\widehat{F}_{t}(Z_{t+1})\geq \widehat{F}_{t}(X_{t})\). The author proves a convergence result in terms of the probabilities \(P(X_{t}\in B_{\epsilon })\), where \(\epsilon \) is a (small) positive number and \( B_{\epsilon }=\{y\in D:F_{0}(y)\leq F_{0}^{\ast }+\epsilon \}\), with \( F_{0}^{\ast }=inf_{y\in D}F_{0}(y)\). He finally gives estimates of the probabilities of entry or leaving the set \(B_{\epsilon }\) of \(\epsilon \) -optimal points from a point \(X_{t}=x_{t}\) lying outside or inside \( B_{\epsilon }\).
The book describes with great details how to solve optimization problems with uncertainty, focusing on convergence results. Many examples illustrate the results.
Reviewer: Alain Brillard (Riedisheim)An optimal control problem with terminal stochastic linear complementarity constraintshttps://zbmath.org/1527.490032024-02-28T19:32:02.718555Z"Luo, Jianfeng"https://zbmath.org/authors/?q=ai:luo.jianfeng"Chen, Xiaojun"https://zbmath.org/authors/?q=ai:chen.xiaojun|chen.xiaojun.1Summary: In this paper, we investigate an optimal control problem with a crucial ODE constraint involving a terminal stochastic LCP and its discrete approximation using the relaxation, the sample average approximation (SAA), and the implicit Euler time-stepping scheme. We show the existence of feasible solutions and optimal solutions to the optimal control problem and its discrete approximation under the condition that the expectation of the stochastic matrix in the stochastic LCP is a Z-matrix or an adequate matrix. Moreover, we prove that the solution sequence generated by the discrete approximation converges to a solution of the original optimal control problem with probability 1 by the repeated limits in the order of \(\epsilon\downarrow 0\), \(\nu\rightarrow\infty\), and \(h\downarrow 0\), where \(\epsilon\) is the relaxation parameter, \(\nu\) is the sample size, and \(h\) is the mesh size. We also provide asymptotics of the SAA optimal value and error bounds of the time-stepping method. A numerical example is used to illustrate the existence of optimal solutions, the discretization scheme, and error estimation.Approximate solution of the optimal control problem for non-linear differential inclusion on the semi-axeshttps://zbmath.org/1527.490042024-02-28T19:32:02.718555Z"Kasimova, Nina"https://zbmath.org/authors/?q=ai:kasimova.nina-v"Zhuk, Tetiana"https://zbmath.org/authors/?q=ai:zhuk.tetiana"Tsyganivska, Iryna"https://zbmath.org/authors/?q=ai:tsyganivska.iryna-mSummary: We consider the optimal control problem of a non-linear system of differential inclusions with fast-oscillating parameters on semi-axes. Using the averaging method, we find an approximate solution for the optimal control of non-linear differential inclusions with fast-oscillating coefficients on a semi-axes. Thus, we prove the convergence of the optimal controls of the initial control problem to the optimal process of the averaged problem and the convergence of the corresponding cost functionals.On I. Meghea and C. S. Stamin review article ``Remarks on some variants of minimal point theorem and Ekeland variational principle with applications''https://zbmath.org/1527.490052024-02-28T19:32:02.718555Z"Göpfert, Alfred"https://zbmath.org/authors/?q=ai:gopfert.alfred"Tammer, Christiane"https://zbmath.org/authors/?q=ai:tammer.christiane"Zălinescu, Constantin"https://zbmath.org/authors/?q=ai:zalinescu.constantinSummary: Being informed that one of our articles is cited in the paper mentioned in the title, we downloaded it, and we were surprised to see that, practically, all the results from our paper were reproduced in Section 3 of \textit{I. Meghea} and \textit{C. S. Stamin}'s article [Demonstr. Math. 55, 354--379 (2022; Zbl 1495.49006)]. Having in view the title of the article, one is tempted to think that the remarks mentioned in the paper are original and there are examples given as to where and how (at least) some of the reviewed results are effectively applied. Unfortunately, a closer look shows that most of those remarks in Section 3 are, in fact, extracted from our article, and it is not shown how a specific result is used in a certain application. So, our aim in the present note is to discuss the content of Section 3 of Meghea and Stamin's paper, emphasizing their Remark 8, in which it is asserted that the proof of Lemma 7 in our article is ``full of errors.''Successive Chebyshev pseudospectral convex optimization method for nonlinear optimal control problemshttps://zbmath.org/1527.490062024-02-28T19:32:02.718555Z"Li, Yang"https://zbmath.org/authors/?q=ai:li.yang.44|li.yang|li.yang.55|li.yang.12|li.yang.13|yang.li.2|li.yang.27|li.yang.11|li.yang.5|li.yang.17|li.yang.2|li.yang.7|li.yang.9|li.yang.6|li.yang.45|li.yang.47|li.yang.40|li.yang.28|li.yang.4|li.yang.8"Chen, Wanchun"https://zbmath.org/authors/?q=ai:chen.wanchun"Yang, Liang"https://zbmath.org/authors/?q=ai:yang.liang.2Summary: This article aims at proposing a successive Chebyshev pseudospectral convex optimization method for solving general nonlinear optimal control problems (OCPs). First, Chebyshev pseudospectral discrete scheme is used to discretize a general nonlinear OCP. At the same time, a convex subproblem is formulated by using the first-order Taylor expansion to convexify the discretized nonlinear dynamic constraints. Second, a trust-region penalty term is added to the performance index of the subproblem, and a successive convex optimization algorithm is proposed to solve the subproblem iteratively. Noted that the trust-region penalty parameters can be adjusted according to the linearization error in iterative process, which improves convergence rate. Third, the Karush-Kuhn-Tucker conditions of the subproblem are derived, and furthermore, a proof is given to show that the algorithm will iteratively converge to the subproblem. Additionally, the global convergence of the algorithm is analyzed and proved, which is based on three key lemmas. Finally, the orbit transfer problem of spacecraft is used to test the performance of the proposed method. The simulation results demonstrate the optimal control is bang-bang form, which is consistent with the result of theoretical proof. Also, the algorithm is of efficiency, fast convergence rate, and high accuracy. Therefore, the proposed method provides a new approach for solving nonlinear OCPs online and has great potential in engineering practice.
{{\copyright} 2021 John Wiley \& Sons Ltd.}A note on the equivalence and the boundary behavior of a class of Sobolev capacitieshttps://zbmath.org/1527.490072024-02-28T19:32:02.718555Z"Christof, Constantin"https://zbmath.org/authors/?q=ai:christof.constantin"Müller, Georg"https://zbmath.org/authors/?q=ai:muller.georg.1|muller.georgSummary: The purpose of this paper is to study different notions of Sobolev capacity commonly used in the analysis of obstacle- and Signorini-type variational inequalities. We review basic facts from capacity theory in an abstract setting that is tailored to the study of \(W^{1,p}\)- and \(W^{1-1}/^{p,p}\)-capacities, and we prove equivalency results that relate several approaches found in the literature to each other. Motivated by applications in contact mechanics, we especially focus on the behavior of different Sobolev capacities on and near the boundary of the domain in question. As a result, we obtain, for example, that the most common approaches to the sensitivity analysis of Signorini-type problems are exactly the same.An analysis on the approximate controllability results for Caputo fractional hemivariational inequalities of order \(1 < r < 2\) using sectorial operatorshttps://zbmath.org/1527.490082024-02-28T19:32:02.718555Z"Raja, Marimuthu Mohan"https://zbmath.org/authors/?q=ai:raja.marimuthu-mohan"Vijayakumar, Velusamy"https://zbmath.org/authors/?q=ai:vijayakumar.velusamy"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-jose"Panda, Sumati Kumari"https://zbmath.org/authors/?q=ai:panda.sumati-kumari"Shukla, Anurag"https://zbmath.org/authors/?q=ai:shukla.anurag"Nisar, Kottakkaran Sooppy"https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaranSummary: In this paper, we investigate the effect of hemivariational inequalities on the approximate controllability of Caputo fractional differential systems. The main results of this study are tested by using multivalued maps, sectorial operators of type \((P, \eta, r, \gamma)\), fractional calculus, and the fixed point theorem. Initially, we introduce the idea of mild solution for fractional hemivariational inequalities. Next, the approximate controllability results of semilinear control problems were then established. Moreover, we will move on to the system involving nonlocal conditions. Finally, an example is provided in support of the main results we acquired.Radial extension of \(\Gamma\)-limitshttps://zbmath.org/1527.490092024-02-28T19:32:02.718555Z"Anza Hafsa, Omar"https://zbmath.org/authors/?q=ai:anza-hafsa.omar"Mandallena, Jean-Philippe"https://zbmath.org/authors/?q=ai:mandallena.jean-philippeA function \(F:X\rightarrow \lbrack 0,\infty ]\), where \(X\) is a vector space, is called radially uniformly upper semicontinuous if there exists \(a>0\) such that \(\lim \sup_{t\rightarrow 1^{-}}\sup_{u\in \mathrm{dom}(F)}\frac{F(tu)-F(u}{a+F(u) }\leq 0\), where \(\mathrm{dom}(F)\) is the effective domain of \(F\).\ The radial extension \(\widehat{F}\) of \(F\) is defined as \(\widehat{F}(u)=\lim \inf_{t\rightarrow 1^{-}}F(tu)\).\ The authors recall properties of radially uniformly upper semicontinuous functions and of their radially extensions, especially a stability result under \(\Gamma \)-convergence:\ If \( \{F_{\varepsilon }\}_{\varepsilon >0}\) is a sequence of radially uniformly upper semicontinuous functions which \(\Gamma \)-converges to \(F\), then \(F\) is radially uniformly upper semicontinuous. If moreover \(\mathrm{dom}(F)\) is super-strongly star-shaped, that is \(t\overline{\mathrm{dom}(F)}\subset \mathrm{int}(\mathrm{dom}(F))\) for all \(t\in \lbrack 0,1]\), then \(\widehat{F}=F\). The authors partly refer to their previous work [Boll. Unione Mat. Ital. 7, No. 1, 1-18 (2014; Zbl 1301.49031)].
The first main result of the paper proves that if \( \{F_{\varepsilon }\}_{\varepsilon >0}\) is a sequence of radially uniformly upper semicontinuous functions such that \(\Gamma \)-\(\lim_{\varepsilon \rightarrow 0}F_{\varepsilon }(u)=F(u)\) on \(D\), and if \(D,E\) satisfy \( tE\subset D\) for all \(t\in \lbrack 0,1]\) and \(\mathrm{dom}(\Gamma \)-\(\lim \inf_{\varepsilon \rightarrow 0}F_{\varepsilon })\subset E\), then \(\Gamma \)-\( \lim_{\varepsilon \rightarrow 0}F_{\varepsilon }(u)=\widehat{F}+\chi _{E}\). The second main result proves that if \(D\subset X\) is super-strongly star-shaped, that is \(t\overline{D}\subset \mathrm{int}D\) for all \(t\in \lbrack 0,1]\) and if \(\{F_{\varepsilon }\}_{\varepsilon >0}\) is a sequence of radially uniformly upper semi-continuous functions such that \(\Gamma \)-\( \lim_{\varepsilon \rightarrow 0}F_{\varepsilon }=F\), then \(\Gamma \)-\( \lim_{\varepsilon \rightarrow 0}(F_{\varepsilon }+\chi _{D})=F+\chi _{ \overline{D}}\). For the proof of these results, the authors mainly use properties of radially uniformly upper semicontinuous functions and of the \( \Gamma \)-convergence.\ The paper ends with an application of such results to homogenization problems with constraints. The authors consider a bounded and open subset \(\Omega \subset \mathbb{R}^{N}\), \(N\geq 1\), the space \(\mathbb{M} \) of \(m\times N\) matrices, \(m\geq 1\), and the space \(X=W^{1,p}(\Omega ; \mathbb{R}^{m})\) equipped with the \(L^{p}\)-norm, \(p>1\). For a Borel measurable function \(f:\Omega \times \mathbb{M}\rightarrow \lbrack 0,\infty ] \), they define a notion of radially uniformly upper semicontinuity which implies that of the functional defined on \(X\) through \(F(u)=\int_{\Omega }f(x,\nabla u(x))dx\). Assuming that \(f:\mathbb{R}^{N}\times \mathbb{M} \rightarrow \lbrack 0,\infty ]\) is a Borel measurable function which is radially uniformly upper semicontinuous, 1-periodic with respect to its first variable and satisfies uniform bounds with respect to the second variable which involve a Borel measurable function \(g\), the authors define \( f_{\varepsilon }(x,\xi )=f(\frac{x}{\varepsilon },\xi)\) and the functional \( F_{\varepsilon }=W^{1,p}(\Omega ;\mathbb{R}^{N})\rightarrow \lbrack 0,\infty ]\) through \(F_{\varepsilon }(u)=\int_{\Omega }f(\frac{x}{\varepsilon } ,\nabla u(x))dx\).
The third main result describes the \(\Gamma \)-limit of \( F_{\varepsilon }\) or of \(F_{\varepsilon }+\chi _{D}\), under different hypotheses on the function \(g\). For the proof, the authors first describe the \(\Gamma \)-convergence of sequences of functionals defined through \( F_{\varepsilon }(u)=\int_{\Omega }f_{\varepsilon }(x,\nabla u(x))dx\), where \( f_{\varepsilon }\) are radially uniformly upper semicontinuous functions from \(\Omega \times \mathbb{M}\) to \([0,\infty ]\), which are not necessarily periodic, but which satisfy bounds involving \(g\). The \(\Gamma \)-\(\lim \sup \) and \(\Gamma \)-\(\lim \inf \) involve the localized quantities \(\mathcal{H}^{\rho }[f](x,\xi )=\inf \left\{ \frac{1}{\left\vert Q_{\rho }(x)\right\vert } \int_{Q_{\rho }(x)}f(y,\xi +\nabla v(y))dy:v\in W_{0}^{1,p}(Q_{\rho }(x); \mathbb{R}^{m})\right\} \) defined for a function \(f:\Omega \times \mathbb{M} \rightarrow \lbrack 0,\infty ]\), where \(Q_{\rho }(x)=x+[-\frac{\rho }{2}, \frac{\rho }{2}]^{N}\).
Reviewer: Alain Brillard (Riedisheim)Gradient damage models for heterogeneous materialshttps://zbmath.org/1527.490102024-02-28T19:32:02.718555Z"Bach, Annika"https://zbmath.org/authors/?q=ai:bach.annika"Esposito, Teresa"https://zbmath.org/authors/?q=ai:esposito.teresa"Marziani, Roberta"https://zbmath.org/authors/?q=ai:marziani.roberta"Zeppieri, Caterina Ida"https://zbmath.org/authors/?q=ai:zeppieri.caterina-idaIn this very nice paper, it is studied, by means of \(\Gamma\)-convergence, a static gradient damage model for periodically heterogeneous elastic materials, modelled through a vanishing parameter \(\delta_\varepsilon\), as \(\varepsilon\) decreases to \(0\). The functional under examination depends on two \(H^1\) fields \(u\) and \(v\), the first representing the deformation and the second, with range \([0,1]\), representing an internal variable v measuring at each point the damage state of the material (the value v =1 corresponding to the original sound state and the value v = 0 corresponding to the totally damaged state). The functional, of integral type, depends also explicitly on \(\varepsilon\) itself and another vanishing (as \(\varepsilon\) decreases to \(0\)), parameter \(\eta_\varepsilon>0\). It generalizes the so called Ambrosio-Tortorelli one, and it is composed of three terms.The first term is the stored elastic energy and reflects the worsening of the elastic properties of the material due to the damage process. Namely, in the regions where the damage occurs, that is, where \(v\) gets close to \(0\), the deformation gradient \(\nabla u\) has a very large norm, namely \(u\) becomes singular. The second term, modelled through a density \((1-v)^2\) represents the energy dissipated in the damage process, and maximizes in the totally damaged state. The third term, penalizes the spatial variations of \(v\), hence, together with the second one, it forces the damage to localize for small \(\varepsilon\) in diffuse regions of size proportional to \(\varepsilon\) , around the set where \(| \nabla u|\) blows up. Then, asymptotically, the damage localization gives rise to sharp cracks and the phase-field functionals, under examination, are expected to behave, in the limit, as fracture models. Three asymptotic regimes (obtained via \(\Gamma\)-convergence with respect to the convergence in measures of the fields \(u\) and \(v\)) are detected according to the limiting ratio \(\frac{\varepsilon}{\delta_\varepsilon}\) as \(\varepsilon \to 0\). The limiting models, generalizing previous works of some of the authors, are composed of two terms, the first one, identical in the three regimes, of bulk type with a `standard' limiting density obtained via classical formulas, and a second one, modelling a surface integrals, requiring a very deep analysis and the employment of many technical tools to be seek, heavily depends upon the limiting ratio \(\frac{\varepsilon}{\delta_\varepsilon}\).
Reviewer: Elvira Zappale (Roma)Interior point methods in optimal control problems of affine systems: convergence results and solving algorithmshttps://zbmath.org/1527.490112024-02-28T19:32:02.718555Z"Malisani, Paul"https://zbmath.org/authors/?q=ai:malisani.paulSummary: This paper presents an interior point method for pure state and mixed-constraint optimal control problems for dynamics, mixed constraints, and cost function all affine in the control variable. This method relies on resolving a sequence of two-point boundary value problems of differential and algebraic equations. This paper establishes a convergence result for primal and dual variables of the optimal control problem. A primal and a primal-dual solving algorithm are presented, and a challenging numerical example is treated for illustration.Codifferentials and quasidifferentials of the expectation of nonsmooth random integrands and two-stage stochastic programminghttps://zbmath.org/1527.490122024-02-28T19:32:02.718555Z"Dolgopolik, M. V."https://zbmath.org/authors/?q=ai:dolgopolik.maxim-vladimirovichSummary: This work is devoted to an analysis of exact penalty functions and optimality conditions for nonsmooth two-stage stochastic programming problems. To this end, we first study the co/quasidifferentiability of the expectation of nonsmooth random integrands and obtain explicit formulae for its co and quasidifferential under some natural assumptions on the integrand. Then, we analyse exact penalty functions for a variational reformulation of two-stage stochastic programming problems and obtain sufficient conditions for the global exactness of these functions with two different penalty terms. In the end of the chapter, we combine our results on the co/quasidifferentiability of the expectation of nonsmooth random integrands and exact penalty functions to derive optimality conditions for nonsmooth two-stage stochastic programming problems in terms of codifferentials.
For the entire collection see [Zbl 1495.90002].Strong variational sufficiency for nonlinear semidefinite programming and its implicationshttps://zbmath.org/1527.490132024-02-28T19:32:02.718555Z"Wang, Shiwei"https://zbmath.org/authors/?q=ai:wang.shiwei"Ding, Chao"https://zbmath.org/authors/?q=ai:ding.chao"Zhang, Yangjing"https://zbmath.org/authors/?q=ai:zhang.yangjing"Zhao, Xinyuan"https://zbmath.org/authors/?q=ai:zhao.xinyuanSummary: Strong variational sufficiency is a newly proposed property, which turns out to be of great use in the convergence analysis of multiplier methods. However, what this property implies for nonpolyhedral problems remains a puzzle. In this paper, we prove the equivalence between the strong variational sufficiency and the strong second-order sufficient condition (SOSC) for nonlinear semidefinite programming (NLSDP) without requiring the uniqueness of the multiplier or any other constraint qualifications. Based on this characterization, the local convergence property of the augmented Lagrangian method (ALM) for NLSDP can be established under the strong SOSC in the absence of constraint qualifications. Moreover, under the strong SOSC, we can apply the semismooth Newton method to solve the ALM subproblems of NLSDP because the positive definiteness of the generalized Hessian of augmented Lagrangian function is satisfied.Exit from singularity. New optimization methods and the \(p\)-regularity theory applicationshttps://zbmath.org/1527.490142024-02-28T19:32:02.718555Z"Evtushenko, Yuri"https://zbmath.org/authors/?q=ai:evtushenko.yuri-g"Malkova, Vlasta"https://zbmath.org/authors/?q=ai:malkova.vlasta"Tret'yakov, Alexey"https://zbmath.org/authors/?q=ai:tretyakov.alexey-aSummary: In the paper, we introduce a new nonsingular operator instead of a degenerate operator of the first derivative in a singular case for solving and describing nonregular optimization problems and some problems in calculus. Such operator is called \(p\)-factor-operator and its construction is based on the derivatives up to order \(p\) as well as on some element \(h\), which we call the ``exit from singularity''. The special variant of the method of the Modified Lagrangian Functions for optimization problems with inequality constraints is justified on the basis of the 2-factor transformation and constructions of \(p\)-regularity theory. These results are used in some classical branches of calculus: implicit function theorem is given for the singular case and is shown the existence of solutions to a boundary-valued problem for a nonlinear differential equation in the resonance case. New numerical methods are proposed including the \(p\)-factor method for solving ODEs with a small parameter and new formula is obtained for the solutions of such type equations.
For the entire collection see [Zbl 1508.90001].Epsilon-regularity for Griffith almost-minimizers in any dimension under a separating conditionhttps://zbmath.org/1527.490152024-02-28T19:32:02.718555Z"Labourie, Camille"https://zbmath.org/authors/?q=ai:labourie.camille"Lemenant, Antoine"https://zbmath.org/authors/?q=ai:lemenant.antoineAuthors' abstract: In this paper we prove that if \((u, K)\) is an almost-minimizer of the Griffith functional and \(K\) is \(\varepsilon\)-close to a plane in some ball \(B\subset \mathbb{R}^N\) while separating the ball \(B\) in two big parts, then \(K\) is \(C^{1,\alpha}\) in a slightly smaller ball. Our result contains and generalizes the \(2\) dimensional result of \textit{J.-F. Babadjian} et al. [J. Eur. Math. Soc. (JEMS) 24, No. 7, 2443--2492 (2022; Zbl 1501.49024)], with a different and more sophisticate approach inspired by \textit{A. Lemenant} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 9, No. 2, 351--384 (2010; Zbl 1197.49050); Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 10, No. 3, 561--609 (2011; Zbl 1239.49062)], using also \textit{C. Labourie} [J. Geom. Anal. 31, No. 10, 10024--10135 (2021; Zbl 1482.49013)] in order to adapt a part of the argument to Griffith minimizers.
Reviewer: Patrick Winkert (Berlin)Weakening state constraints in optimal control problemshttps://zbmath.org/1527.490162024-02-28T19:32:02.718555Z"Aseev, S. M."https://zbmath.org/authors/?q=ai:aseev.sergei-mThe author considers and compares two optimal control problems.
Problem (P):
\begin{align*}
&J(T,x(\cdot))=\varphi(T,x(0),x(T))+\int_0^T\lambda(x(t))\delta_M(x(t))\,dt\rightarrow\,\min,\\
&\dot x(t)\in F(x(t));\qquad x(0)\in M_0,\ x(T)\in M_1,
\end{align*}
Problem (Q):
\begin{align*}
&\tilde J(T,x(\cdot))=\varphi(T,x(0),x(T))\rightarrow\,\min,\\
&\dot x(t)\in F(x(t)),\ x(t)\in G,\ t\in [0,T];\qquad x(0)\in M_0,\ x(T)\in M_1.
\end{align*}
The main difference of Problem (P) from Problem (Q) is the weakening of the strict risk aversion condition \(x(t)\in G\). This circumstance makes the statement of Problem (P) more suitable for many applications. Conditions are obtained that guarantee the equivalence of both problems.
Reviewer: Igor Bock (Bratislava)Optimization of the reachable set of a linear system with respect to another sethttps://zbmath.org/1527.490172024-02-28T19:32:02.718555Z"Balashov, M. V."https://zbmath.org/authors/?q=ai:balashov.maxim-viktorovich"Kamalov, R. A."https://zbmath.org/authors/?q=ai:kamalov.rinat-aSummary: Given a linear controlled autonomous system, we consider the problem of including a convex compact set in the reachable set of the system in the minimum time and the problem of determining the maximum time when the reachable set can be included in a convex compact set. Additionally, the initial point and the time at which the extreme time is achieved in each problem are determined. Each problem is discretized on a grid of unit vectors and is then reduced to a linear programming problem to find an approximate solution of the original problem. Additionally, error estimates for the solution are found. The problems are united by a common ideology going back to the problem of finding the Chebyshev center.Optimal geodesic curvature constrained Dubins' paths on a spherehttps://zbmath.org/1527.490182024-02-28T19:32:02.718555Z"Darbha, Swaroop"https://zbmath.org/authors/?q=ai:darbha.swaroop"Pavan, Athindra"https://zbmath.org/authors/?q=ai:pavan.athindra"Kumbakonam, Rajagopal"https://zbmath.org/authors/?q=ai:kumbakonam.rajagopal"Rathinam, Sivakumar"https://zbmath.org/authors/?q=ai:rathinam.sivakumar"Casbeer, David W."https://zbmath.org/authors/?q=ai:casbeer.david-wellman"Manyam, Satyanarayana G."https://zbmath.org/authors/?q=ai:manyam.satyanarayana-guptaSummary: In this article, we consider the motion planning of a rigid object on the unit sphere with a unit speed. The motion of the object is constrained by the maximum absolute value, \(U_{max}\), of geodesic curvature of its path; this constrains the object to change the heading at the fastest rate only when traveling on a tight smaller circular arc of radius \(r <1\), where \(r\) depends on the bound, \(U_{max}\). We show in this article that if \(0< r \leq \frac{1}{2}\), the shortest path between any two configurations of the rigid body on the sphere consists of a concatenation of at most three circular arcs. Specifically, if \(C\) is the smaller circular arc and \(G\) is the great circular arc, then the optimal path can only be \(CCC\), \(CGC\), \(CC\), \(CG\), \(GC\), \(C\) or \(G\). If \(r>\frac{1}{2}\), while paths of the above type may cease to exist depending on the boundary conditions and the value of \(r\), optimal paths may be concatenations of more than three circular arcs.Extremal trajectories in a time-optimal problem on the group of motions of a plane with admissible control in a circular sectorhttps://zbmath.org/1527.490192024-02-28T19:32:02.718555Z"Mashtakov, Alexey P."https://zbmath.org/authors/?q=ai:mashtakov.alexey-p"Sachkov, Yuri L."https://zbmath.org/authors/?q=ai:sachkov.yuri-lThe authors study the time-optimal problem of a car that moves on a plane and has a minimum turning radius proving the controllability and existence of optimal trajectories. They deduce a Hamiltonian system for extremals by using optimality conditions based on the Pontryagin maximum principle. Qualitative study of the Hamiltonian and explicit expressions for the extremal control and trajectories are derived. It is shown that the studied system can be used for detections of salient lines in image processing.
Reviewer: Liviu Goraş (Iaşi)Optimal control problems without terminal constraints: the turnpike property with interior decayhttps://zbmath.org/1527.490202024-02-28T19:32:02.718555Z"Gugat, Martin"https://zbmath.org/authors/?q=ai:gugat.martin"Lazar, Martin"https://zbmath.org/authors/?q=ai:lazar.martinSummary: We show a turnpike result for problems of optimal control with possibly nonlinear systems as well as pointwise-in-time state and control constraints. The objective functional is of integral type and contains a tracking term which penalizes the distance to a desired steady state. In the optimal control problem, only the initial state is prescribed. We assume that a cheap control condition holds that yields a bound for the optimal value of our optimal control problem in terms of the initial data. We show that the solutions to the optimal control problems on the time intervals \([0,T]\) have a turnpike structure in the following sense: For large \(T\) the contribution to the objective functional that comes from the subinterval \([T/2,T]\), i.e., from the second half of the time interval \([0,T]\), is at most of the order \(1/T\). More generally, the result holds for subintervals of the form \([rT,T]\), where \(r\in (0,1/2)\) is a real number. Using this result inductively implies that the decay of the integral on such a subinterval in the objective function is faster than the reciprocal value of a power series in \(T\) with positive coefficients. Accordingly, the contribution to the objective value from the final part of the time interval decays rapidly with a growing time horizon. At the end of the paper we present examples for optimal control problems where our results are applicable.A control space ensuring the strong convergence of continuous approximation for a controlled sweeping processhttps://zbmath.org/1527.490212024-02-28T19:32:02.718555Z"Nour, Chadi"https://zbmath.org/authors/?q=ai:nour.chadi"Zeidan, Vera"https://zbmath.org/authors/?q=ai:zeidan.vera-michelRecall that the classic Moreau's sweeping process is modeled by some differential equation that describes the motion of a point under the influence of a moving set. That equation is a non-smooth differential equation with its right-hand side not continuous. To solve the equation is challenging, but it is also a very important in practice. The mentioned equation can be used to model a wide variety of phenomena, including elastoplasticity, crowd motion, and hysteresis.
In this paper the following fixed time Mayer-type optimal control problem (P) involving \(W^{1,2}\)-controlled sweeping systems is considered:
\[
\begin{aligned}
(P) &\quad \text{Minimize} \quad g(x(0), x(1)) \\
&\quad \text{over} \quad (x, u) \in \text{AC}([0, 1]; \mathbb{R}^n) \times \mathcal{W}
\end{aligned}
\]
such that
\[
\begin{cases} \quad \dot{x}(t) \in f(x(t), u(t)) - \partial \phi(x(t)), \text{ a.e. } t \in [0, 1], \\
\quad x(0) \in C_0 \subset \text{dom } \phi, \\
\quad x(1) \in C_1. \end{cases}
\]
where, \( g : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R} \cup \{ \infty \}, \) \( f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n, \) \( \phi : \mathbb{R}^n \to \mathbb{R} \cup \{ \infty \}, \) \( \partial \) denotes the Clarke subdifferential, \( C := \text{dom} \;\phi \) is the zero-sublevel set of a function \( \psi : \mathbb{R}^n \to \mathbb{R} \), that is, \( C = \{ x \in \mathbb{R}^n : \psi(x) \le 0 \}, \) \( C_0 \subset C, \) \( C_1 \subset \mathbb{R}^n \) and for a multifunction \( U : [0, 1] \rightrightarrows \mathbb{R}^m \) and \( U := \bigcup_{t \in [0, 1]} U(t) \), the set of control functions \( \mathcal{W}:= W^{1,2}([0, 1];U)\).
The authors establish existence of optimal solutions and derive local optimality conditions in the form of a weak-Pontryagin-type maximum principle via strong converging continuous approximations, whose state entirely resides in the interior of the prox-regular set. Subdifferentials smaller than the standard ones are employed in the optimality results. An example of the problem (P) demonstrating the usage the obtained necessary optimality conditions is given. The paper contains on ample references consisting of 40 items.
Reviewer: Wiesław Kotarski (Sosnowiec)Fréchet second-order subdifferentials of Lagrangian functions and optimality conditionshttps://zbmath.org/1527.490222024-02-28T19:32:02.718555Z"An, Duong Thi Viet"https://zbmath.org/authors/?q=ai:viet-an.duong-thi"Xu, Hong-Kun"https://zbmath.org/authors/?q=ai:xu.hong-kun"Yen, Nguyen Dong"https://zbmath.org/authors/?q=ai:yen.nguyen-dongThis paper address minimization problems with abstract constraints in infinite-dimensional spaces. The primary focus of this research is to establish results concerning second-order optimality conditions. Notably, the study assumes that the objective function involved is only \(C^1\)-smooth.
More precisely, the authors consider the minimization problem \(\min\{\varphi(x)\ \mid\ x\in X \text{ and } G(x)\in K\}\), where \(\varphi:X\to\mathbb{R}\) and \(G:X\to Y\) are \(C^1\)-smooth functions, \(X\) and \(Y\) are Banach spaces, and \(K\subset Y\) is a closed convex cone.
Second order necessary optimality conditions are given when \(K=\{0\}\) or when \(G\) is an affine operator, and sufficient optimality conditions are given when \(K=\{0\}\) and \(X\) is a Hilbert space (or reflexive Banach space).
One of the key strengths of this work is the application of the concept of Fréchet (regular) second-order subdifferential from variational analysis to the Lagrangian function of the problem.
Reviewer: Jérôme Lohéac (Vandœuvre-lès-Nancy)On Integer optimal control with total variation regularization on multidimensional domainshttps://zbmath.org/1527.490232024-02-28T19:32:02.718555Z"Manns, Paul"https://zbmath.org/authors/?q=ai:manns.paul"Schiemann, Annika"https://zbmath.org/authors/?q=ai:schiemann.annikaSummary: We consider optimal control problems with integer-valued controls and a total variation regularization penalty in the objective on domains of dimension two or higher. The penalty yields that the feasible set is sequentially closed in the weak-\(^*\) topology and closed in the strict topology in the space of functions of bounded variation. In turn, we derive first-order optimality conditions of the optimal control problem as well as trust-region subproblems with partially linearized model functions using local variations of the level sets of the feasible control functions. We also prove that a recently proposed function space trust-region algorithm -- sequential linear integer programming -- produces sequences of iterates whose limits are first-order optimal points.Numerical solution of delay fractional optimal control problems with free terminal timehttps://zbmath.org/1527.490242024-02-28T19:32:02.718555Z"Liu, Chongyang"https://zbmath.org/authors/?q=ai:liu.chongyang"Gong, Zhaohua"https://zbmath.org/authors/?q=ai:gong.zhaohua"Wang, Song"https://zbmath.org/authors/?q=ai:wang.song.2"Teo, Kok Lay"https://zbmath.org/authors/?q=ai:teo.kok-laySummary: This paper considers a class of delay fractional optimal control problems with free terminal time. The fractional derivatives in this class of problems are described in the Caputo sense and they can be of different orders. We first show that for this class of problems, the well-known time-scaling transformation for mapping the free time horizon into a fixed time interval yields a new fractional-order system with variable time-delay. Then, we propose an explicit numerical scheme for solving the resulting fractional time-delay system, which gives rise to a discrete-time optimal control problem. Furthermore, we derive gradient formulas of the cost and constraint functions with respect to decision variables. On this basis, a gradient-based optimization approach is developed to solve the resulting discrete-time optimal control problem. Finally, an example problems is solved to demonstrate the effectiveness of our proposed solution approach.Moments and convex optimization for analysis and control of nonlinear PDEshttps://zbmath.org/1527.490252024-02-28T19:32:02.718555Z"Korda, Milan"https://zbmath.org/authors/?q=ai:korda.milan"Henrion, Didier"https://zbmath.org/authors/?q=ai:henrion.didier"Lasserre, Jean Bernard"https://zbmath.org/authors/?q=ai:lasserre.jean-bernardSummary: This work presents a convex-optimization-based framework for analysis and control of nonlinear partial differential equations. A measure-valued approach uses a particular weak embedding of the nonlinear PDE and results in \textit{linear} equations in appropriate space of Borel measures. These equations are then used as linear constraints of an infinite-dimensional linear programming problem (LP). This LP is then approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems (SDPs). For analysis of uncontrolled PDEs, optimal solutions to these SDPs provide bounds on any specified polynomial functional integral of the solutions to the PDE. For PDE control, optimal solutions to these SDPs provide bounds on the optimal value of a given optimal control problem as well as suboptimal feedback controllers. The entire approach is based purely on convex optimization and is mesh-free, i.e., does not rely on spatio-temporal gridding, even though the PDE addressed can be fully nonlinear. It is applicable to a very broad class of nonlinear PDEs with polynomial data. Computational complexity is analyzed and several complexity reduction procedures are described. Numerical examples demonstrate the approach.
For the entire collection see [Zbl 1492.49003].Fractional semilinear optimal control: optimality conditions, convergence, and error analysishttps://zbmath.org/1527.490262024-02-28T19:32:02.718555Z"Otárola, Enrique"https://zbmath.org/authors/?q=ai:otarola.enriqueSummary: We adopt the integral definition of the fractional Laplace operator and analyze an optimal control problem for a fractional semilinear elliptic partial differential equation (PDE); control constraints are also considered. We establish the well-posedness of fractional semilinear elliptic PDEs and analyze regularity properties and suitable finite element discretizations. Within the setting of our optimal control problem, we derive the existence of optimal solutions as well as first and second order optimality conditions; regularity estimates for the optimal variables are also analyzed. We devise a fully discrete scheme that approximates the control variable with piecewise constant functions; the state and adjoint equations are discretized with continuous piecewise linear finite elements. We analyze convergence properties of discretizations and derive a priori error estimates.Hölder continuity of the minimizer of an obstacle problem with generalized Orlicz growthhttps://zbmath.org/1527.490272024-02-28T19:32:02.718555Z"Karppinen, Arttu"https://zbmath.org/authors/?q=ai:karppinen.arttu"Lee, Mikyoung"https://zbmath.org/authors/?q=ai:lee.mikyoungIn this paper, the authors investigate on Hölder continuity of the minimizer of an obstacle problem with generalized Orlicz growth. More precisely, the derived results cover as special cases standard, variable exponent, double phase, and Orlicz growth.
Reviewer: Savin Treanţă (Bucureşti)Pursuit differential game described by infinite first order 2-systems of differential equationshttps://zbmath.org/1527.490282024-02-28T19:32:02.718555Z"Ibragimov, G."https://zbmath.org/authors/?q=ai:ibragimov.gafurjan-i|ibragimov.g-l|ibragimov.g-t"Akhmedov, A."https://zbmath.org/authors/?q=ai:akhmedov.anar|akhmedov.askar|ahmedov.anvarjon-a|akhmedov.ali-m|akhmedov.azer|akhmedov.a-a|akhmedov.a-sh|akhmedov.a-i|akhmedov.akhmed|akhmedov.anvarjon|akhmedov.a-b"Puteri Nur Izzati"https://zbmath.org/authors/?q=ai:puteri-nur-izzati."Abdul Manaf, N."https://zbmath.org/authors/?q=ai:abdul-manaf.nSummary: We study a pursuit differential game problem for infinite first order 2-systems of differential equations in the Hilbert space \(l_2\). Geometric constraints are imposed on controls of players. If the state of system coincides with the origin, then we say that pursuit is completed. In the game, pursuer tries to complete the game, while the aim of evader is opposite. The problem is to find a formula for guaranteed pursuit time. In the present paper, an equation for guaranteed pursuit time is obtained. Moreover, a strategy for the pursuer is constructed in explicit form. To prove the main result, we use solution of a control problem.Optimal control problems in transport dynamics with additive noisehttps://zbmath.org/1527.490292024-02-28T19:32:02.718555Z"Almi, Stefano"https://zbmath.org/authors/?q=ai:almi.stefano"Morandotti, Marco"https://zbmath.org/authors/?q=ai:morandotti.marco"Solombrino, Francesco"https://zbmath.org/authors/?q=ai:solombrino.francescoMotivated by the applications in leader-follower multi-agent dynamics, the authors study a class of optimal control problems, where the goal is to influence the behavior of a given population through another controlled one interacting with the first. The evolution of the systems obeys a non-linear Fokker-Planck-type equation as diffusive terms accounting for randomness in the evolution are taken into account. In particular, the authors extend the work by \textit{M. Bongini} and \textit{G. Buttazzo} [Math. Models Methods Appl. Sci. 27, No. 3, 427--451 (2017; Zbl 1365.49004)]. A well-posedness theory under very low regularity of the control vector fields is developed. The mean-field limit of a stochastic particle approximation is also detailed.
Reviewer: A. Omrane (Cayenne)Discrete potential mean field games: duality and numerical resolutionhttps://zbmath.org/1527.490302024-02-28T19:32:02.718555Z"Bonnans, J. Frédéric"https://zbmath.org/authors/?q=ai:bonnans.joseph-frederic"Lavigne, Pierre"https://zbmath.org/authors/?q=ai:lavigne.pierre"Pfeiffer, Laurent"https://zbmath.org/authors/?q=ai:pfeiffer.laurentA general class of discrete time and finite state space mean field game problems with potential structure incorporating interactions through a congestion term and a price variable are introduced and studied. Connections between these game problems and two optimal control problems are investigated by meas of duality. Two families of numerical methods, namely primal-dual proximal and alternating direction method of multipliers, are employed for solving such problems, and computational results for two examples with hard constraints are provided.
Reviewer: Sorin-Mihai Grad (Paris)Harmonic intrinsic graphs in the Heisenberg grouphttps://zbmath.org/1527.490312024-02-28T19:32:02.718555Z"Young, Robert"https://zbmath.org/authors/?q=ai:young.robert-m|young.robert-l|young.robert-e|young.robertSummary: Minimal surfaces in \(\mathbb{R}^n\) can be locally approximated by graphs of harmonic functions, \textit{i.e.}, functions that are critical points of the Dirichlet energy, but no analogous theorem is known for \(H\)-minimal surfaces in the three-dimensional Heisenberg group \(\mathbb{H}\), which are known to have singularities. In this paper, we introduce a definition of intrinsic Dirichlet energy for surfaces in \(\mathbb{H}\) and study the critical points of this energy, which we call contact harmonic graphs. Nearly flat regions of \(H\)-minimal surfaces can often be approximated by such graphs. We give a calibration condition for an intrinsic Lipschitz graph to be energy-minimizing, construct energy-minimizing graphs with a variety of singularities, and prove a first variation formula for the energy of intrinsic Lipschitz graphs and piecewise smooth intrinsic graphs.On an effective approach in shape optimization problem for Stokes equationhttps://zbmath.org/1527.490322024-02-28T19:32:02.718555Z"Chakib, Abdelkrim"https://zbmath.org/authors/?q=ai:chakib.abdelkrim"Khalil, Ibrahim"https://zbmath.org/authors/?q=ai:khalil.ibrahim"Ouaissa, Hamid"https://zbmath.org/authors/?q=ai:ouaissa.hamid"Sadik, Azeddine"https://zbmath.org/authors/?q=ai:sadik.azeddineSummary: In this paper, we deal with the numerical study of a shape optimization problem governed by Stokes system. More precisely, we propose an effective numerical approach based on the shape derivative formula with respect to convex domains using Minkowski deformation [\textit{A. Boulkhemair} and \textit{A. Chakib}, J. Convex Anal. 21, No. 1, 67--87 (2014; Zbl 1290.49084)]. Then, we present some numerical tests including comparison results showing that the proposed algorithm is more efficient, in term of the accuracy of the solution and central processing unit (CPU) time execution, than the one involving the classical shape derivative formula massively used in literature.The strip method for shape derivativeshttps://zbmath.org/1527.490332024-02-28T19:32:02.718555Z"Hardesty, Sean"https://zbmath.org/authors/?q=ai:hardesty.sean"Antil, Harbir"https://zbmath.org/authors/?q=ai:antil.harbir"Kouri, Drew P."https://zbmath.org/authors/?q=ai:kouri.drew-p"Ridzal, Denis"https://zbmath.org/authors/?q=ai:ridzal.denisSummary: A major challenge in shape optimization is the coupling of finite element method (FEM) codes in a way that facilitates efficient computation of shape derivatives. This is particularly difficult with multiphysics problems involving legacy codes, where the costs of implementing and maintaining shape derivative capabilities are prohibitive. The volume and boundary methods are two approaches to computing shape derivatives. Each has a major drawback: the boundary method is less accurate, while the volume method is more invasive to the FEM code. We introduce the \textit{strip method}, which computes shape derivatives on a strip adjacent to the boundary. The strip method makes code coupling simple. Like the boundary method, it queries the state and adjoint solutions at quadrature nodes, but requires no knowledge of the FEM code implementations. At the same time, it exhibits the higher accuracy of the volume method. As an added benefit, its computational complexity is comparable to that of the boundary method, that is, it is faster than the volume method. We illustrate the benefits of the strip method with numerical examples.
{{\copyright} 2021 John Wiley \& Sons, Ltd.}Uniform bound on the number of partitions for optimal configurations of the Ohta-Kawasaki energy in 3Dhttps://zbmath.org/1527.490342024-02-28T19:32:02.718555Z"Lu, Xin Yang"https://zbmath.org/authors/?q=ai:lu.xinyang|lu.xinyang.1"Wei, Jun-Cheng"https://zbmath.org/authors/?q=ai:wei.junchengSummary: We study a 3D ternary system which combines an interface energy with a long-range interaction term. Several such systems were derived as a sharp-interface limit of the Nakazawa-Ohta density functional theory of triblock copolymers. Both the binary case in 2D and 3D, and the ternary case in 2D, are quite well understood, whereas very little is known about the ternary case in 3D. In particular, it is even unclear whether minimizers are made of finitely many components. In this paper, we provide a positive answer to this, by proving that the number of components in a minimizer is bounded from above by a computable quantity depending only on the total masses and the interaction coefficients. There are two key difficulties, namely, the impossibility to decouple the long-range interaction from the perimeter term, and the absence of a quantitative isoperimetric inequality with two mass constraints in 3D. Therefore, the actual shape of minimizers is unknown, even for small masses, making the construction of suitable competing configurations significantly more delicate.Shape optimization method for strength design problem of microstructures in a multiscale structurehttps://zbmath.org/1527.490352024-02-28T19:32:02.718555Z"Torisaki, Mihiro"https://zbmath.org/authors/?q=ai:torisaki.mihiro"Shimoda, Masatoshi"https://zbmath.org/authors/?q=ai:shimoda.masatoshi"Al Ali, Musaddiq"https://zbmath.org/authors/?q=ai:al-ali.musaddiqSummary: In this paper, we propose a solution to a shape optimization problem for the strength design of periodic microstructures in multiscale structures. Two maximum stress minimization problems are addressed: minimization of maximum microstructural stress and minimization of maximum macrostructural stress. The homogenization method is used to bridge the macrostructure and the microstructure and to calculate local microstructural stress. By replacing the maximum stress value with a Kreisselmeier-Steinhauser function, the difficulty of nondifferentiability of maximum stress is avoided. Each strength design problem is formulated as a distributed parameter optimization problem subject to an area constraint including the whole microstructure. The shape gradient functions for both problems are derived using Lagrange's undetermined multiplier method, the material derivative method, and the adjoint variable method. The \(\mathrm{H}^1\) gradient method is used to determine the unit cell shapes of the microstructure, while reducing the objective function and maintaining smooth design boundaries. In the numerical examples, the optimal shapes obtained for minimization of the maximum local stress of the microstructure and the macrostructure are compared and discussed. The results confirm the effectiveness of the microstructure shape optimization method for the two strength design problems of multiscale structures.
{{\copyright} 2022 John Wiley \& Sons Ltd.}Boundedness and unboundedness in total variation regularizationhttps://zbmath.org/1527.490362024-02-28T19:32:02.718555Z"Bredies, Kristian"https://zbmath.org/authors/?q=ai:bredies.kristian"Iglesias, José A."https://zbmath.org/authors/?q=ai:iglesias.jose-antonio"Mercier, Gwenael"https://zbmath.org/authors/?q=ai:mercier.gwenaelIn mathematics, the total variation identifies several slightly different concepts, related to the local or global structure of the codomain of a function or a measure. The total variation denoising, also known as total variation regularization or total variation filtering, is a noise removal process (filter). It is based on the principle that signals with excessive and possibly spurious detail have high total variation, that is, the integral of the image gradient magnitude is high. The total variation regularization have had an important role in several classical problems of the geometric measure theory, signal processing and physics.
Several authors studied the total variation regularization [\textit{V. Caselles} et al., Rev. Mat. Iberoam. 27, No. 1, 233--252 (2011; Zbl 1228.94005); \textit{K. Bredies} et al., SIAM J. Imaging Sci. 3, No. 3, 492--526 (2010; Zbl 1195.49025); \textit{A. Chambolle} and \textit{P-L. Lions}, Numer. Math. 76, No. 2, 167--188 (1997; Zbl 0874.68299); \textit{E. Gonzalez} and \textit{U. Massari} Rend. Sem. Mat. Univ. Politec. Torino 52, No. 1, 1--28 (1994; Zbl 0819.49025); \textit{M. Grasmair} and \textit{A. Obereder} Numer. Funct. Anal. Optim. 29, No. 3--4, 346--361 (2008; Zbl 1142.65013); \textit{M. Grasmair}, J. Math. Imaging Vision 27, No. 1, 59--66 (2007; Zbl 1478.94041); \textit{V. Gutev}, J. Math. Anal. Appl. 491, No. 1, Article ID 124242, 12 p. (2020; Zbl 1519.54006); \textit{C. Kirisits} et al., SIAM J. Imaging Sci. 12, No. 4, 1643--1668 (2019; Zbl 1439.49066); \textit{T. Valkonen}, Inverse Probl. 37, No. 4, Article ID 045010, 30 p. (2021; Zbl 1515.65147)].
The principal objective in this paper is to study boundedness and unboundedness in total variation regularization. The authors present a simple proof of boundedness of the minimizer for fixed regularization parameter, and obtain a boundedness result for the case of infimal convolution of first and second order total variation regularizers, for which the optimality conditions are closely related to subgradients of total variation.
Reviewer: Lakehal Belarbi (Mostaganem)Heterogeneous gradient flows in the topology of fibered optimal transporthttps://zbmath.org/1527.490372024-02-28T19:32:02.718555Z"Peszek, Jan"https://zbmath.org/authors/?q=ai:peszek.jan"Poyato, David"https://zbmath.org/authors/?q=ai:poyato.davidSummary: We introduce an optimal transport topology on the space of probability measures over a fiber bundle, which penalizes the transport cost from one fiber to another. For simplicity, we illustrate our construction in the Euclidean case \(\mathbb{R}^d\times\mathbb{R}^d\), where we penalize the quadratic cost in the second component. Optimal transport becomes then constrained to happen along fixed fibers. Despite the degeneracy of the infinitely-valued and discontinuous cost, we prove that the space of probability measures \((\mathcal{P}_{2,\nu}(\mathbb{R}^{2d}), W_{2, \nu})\) with fixed marginal \(\nu\in\mathcal{P}(\mathbb{R}^d)\) in the second component becomes a Polish space under the fibered transport distance, which enjoys a weak Riemannian structure reminiscent of the one proposed by F. Otto for the classical quadratic Wasserstein space. Three fundamental issues are addressed: 1) We develop an abstract theory of gradient flows with respect to the new topology; 2) We show applications that identify a novel fibered gradient flow structure on a large class of evolution PDEs with heterogeneities; 3) We exploit our method to derive long-time behavior and global-in-time mean-field limits in a multidimensional Cucker-Smale-type alignment model with weakly singular coupling.Weighted Minkowski's existence theorem and projection bodieshttps://zbmath.org/1527.520072024-02-28T19:32:02.718555Z"Kryvonos, Liudmyla"https://zbmath.org/authors/?q=ai:kryvonos.liudmyla"Langharst, Dylan"https://zbmath.org/authors/?q=ai:langharst.dylanSummary: The Brunn-Minkowski Theory has seen several generalizations over the past century. Many of the core ideas have been generalized to measures. With the goal of framing these generalizations as a weighted Brunn-Minkowski theory, we prove the Minkowski existence theorem for a large class of Borel measures with continuous density, denoted by \(\Lambda^n\): for \(\nu\) a finite, even Borel measure on the unit sphere and even \(\mu \in \Lambda^n\), there exists a symmetric convex body \(K\) such that
\[d\nu (u)=c_{\mu ,K}dS^{\mu }_K(u),\]
where \(c_{\mu ,K}\) is a quantity that depends on \(\mu\) and \(K\) and \(dS^{\mu }_K(u)\) is the surface area-measure of \(K\) with respect to \(\mu \). Examples of measures in \(\Lambda^n\) are homogeneous measures (with \(c_{\mu ,K}=1)\) and probability measures with radially decreasing densities (e.g. the Gaussian measure). We will also consider weighted projection bodies \(\Pi_\mu K\) by classifying them and studying the isomorphic Shephard problem: if \(\mu\) and \(\nu\) are even, homogeneous measures with density and \(K\) and \(L\) are symmetric convex bodies such that \(\Pi_{\mu } K \subset \Pi_{\nu } L\), then can one find an optimal quantity \(\mathcal{A}>0\) such that \(\mu (K)\leq \mathcal{A}\nu (L)\)? Among other things, we show that, in the case where \(\mu =\nu\) and \(L\) is a projection body, \( \mathcal{A}=1\).On the existence of closed \(C^{1, 1}\) curves of constant curvaturehttps://zbmath.org/1527.530022024-02-28T19:32:02.718555Z"Ketover, Daniel"https://zbmath.org/authors/?q=ai:ketover.daniel"Liokumovich, Yevgeny"https://zbmath.org/authors/?q=ai:liokumovich.yevgenyThis paper establishes a variant of a conjecture of \textit{V. I. Arnold} [Arnold's problems. Translated and revised edition of the 2000 Russian original. Berlin: Springer; Moscow: PHASIS (2004; Zbl 1051.00002)], namely for every positive constant \(c<\infty\) the existence of an immersed \(C^{1,1}\)-curve on any Riemannian surface, which is smooth with curvature equal to \(\pm c\) except possibly at one single point \(p\). The authors provide examples to illustrate that the regularity of the resulting curve cannot be enhanced in general.
Reviewer: Hsiao-Fan Liu (New Taipei)Minimal bubbling for Willmore surfaceshttps://zbmath.org/1527.530092024-02-28T19:32:02.718555Z"Marque, Nicolas"https://zbmath.org/authors/?q=ai:marque.nicolasSummary: In this paper, we build an explicit example of a minimal bubble on a Willmore surface, showing there cannot be compactness for Willmore immersions of Willmore energy above \(16\pi\). Additionally, we prove an inequality on the 2nd residue for limits of sequences of Willmore immersions with simple minimal bubbles. Doing so, we exclude some gluing configurations and prove compactness for immersed Willmore tori of energy below \(12\pi\).New second-order optimality conditions in sub-Riemannian geometryhttps://zbmath.org/1527.530232024-02-28T19:32:02.718555Z"Jóźwikowski, Michał"https://zbmath.org/authors/?q=ai:jozwikowski.michalThe author deals with the smoothness problem for all minimizing sub-Riemannian geodesics. He finds an ordinary differential equation for the velocity of an abnormal sub-Riemannian geodesic. After dividing abnormal minimizers into two classes, called 2-normal and 2-abnormal extremals, the author proposes the proof of regularity of all 2-normal extremals with an open problem of proving the same result also for abnormal extremals.
Reviewer: Igor Bock (Bratislava)Carnot rectifiability of sub-Riemannian manifolds with constant tangenthttps://zbmath.org/1527.530252024-02-28T19:32:02.718555Z"Le Donne, Enrico"https://zbmath.org/authors/?q=ai:le-donne.enrico"Young, Robert"https://zbmath.org/authors/?q=ai:young.robert-l|young.robert-m|young.robert|young.robert-eThe authors propose a new tool for rectifiability Carnot groups. More specifically, they show that if \(M\) is a sub-Riemannian manifold and \(N\) is a Carnot group such that the nilpotentization of \(M\) at almost every point is isomorphic to \(N\), then there are subsets of \(N\) of positive measure that embed into \(M\) by bi-Lipschitz maps. Furthermore, \(M\) is countably \(N\)-rectifiable, i.e., all of \(M\) except for a null set can be covered by countably many such maps.
Reviewer: Peibiao Zhao (Nanjing)Convergence of metric measure spaces satisfying the CD condition for negative values of the dimension parameterhttps://zbmath.org/1527.530382024-02-28T19:32:02.718555Z"Magnabosco, Mattia"https://zbmath.org/authors/?q=ai:magnabosco.mattia"Rigoni, Chiara"https://zbmath.org/authors/?q=ai:rigoni.chiara"Sosa, Gerardo"https://zbmath.org/authors/?q=ai:sosa.gerardoSummary: We study the problem of whether the curvature-dimension condition with negative values of the generalized dimension parameter is stable under a suitable notion of convergence. To this purpose, first of all we propose an appropriate setting to introduce the \(\mathsf{CD} (K, N)\) condition for \(N < 0\), allowing metric measure structures in which the reference measure is quasi-Radon. Then in this class of spaces we define the distance \(\mathsf{d}_{\mathsf{iKRW}}\), which extends the already existing notions of distance between metric measure spaces. Finally, we prove that if a sequence of metric measure spaces satisfying the \(\mathsf{CD} (K,N)\) condition with \(N < 0\) is converging with respect to the distance \(\mathsf{d}_{\mathsf{iKRW}}\) to some metric measure space, then this limit structure is still a \(\mathsf{CD} (K,N)\) space.Constant mean curvature spheres in homogeneous three-manifoldshttps://zbmath.org/1527.530512024-02-28T19:32:02.718555Z"Meeks, William H. III"https://zbmath.org/authors/?q=ai:meeks.william-h-iii"Mira, Pablo"https://zbmath.org/authors/?q=ai:mira.pablo"Pérez, Joaquín"https://zbmath.org/authors/?q=ai:perez.joaquin"Ros, Antonio"https://zbmath.org/authors/?q=ai:ros.antonioThe authors study CMC spheres in a Riemannian homogeneous three-manifold \(M\).
We refer to [\textit{J. W. Milnor}, Adv. Math. 21, 293--329 (1976; Zbl 0341.53030); \textit{K. Sekigawa}, Tensor, New Ser. 31, 87--97 (1977; Zbl 0356.53016); \textit{P. Scott}, Bull. London Math. Soc. 15, 401--487 (1983; Zbl 0561.57001); \textit{B. Daniel}, Comment. Math. Helv. 82, 87--131 (2007; Zbl 1123.53029)] for Riemannian homogeneous three-manifolds. Denote by \(X\) the Riemannian universal cover of \(M\). The isometry group \(I(X)\) of \(X\) has dimension 6, 4 or 3. If \(\dim I(X)=6\), then \(M\) is a space form. It is well-known that any CMC sphere in \(X\) is totally umbilical and therefore the boundary of a geodesic ball. Originally, this was obtained for \(X=\mathbb{R}^3\) (Hopf's theorem). If \(\dim I(X)=4\), then \(X\) has rotational symmetry and it is isometric to one of \(\mathbb{S}^2 (\kappa )\times \mathbb{R}\) (\(\kappa >0\)), \(\mathbb{H}^2 (\kappa )\times \mathbb{R}\) (\(\kappa <0\)), a Berger sphere, the Heisenberg group \(\mathrm{Nil}_3\), or the universal cover \(\widetilde{\mathrm{SL}}(2, \mathbb{R})\) of \(\mathrm{SL}(2, \mathbb{R})\) with a left invariant metric. These are Riemannian fibrations over complete 2-dimensional space forms. The last three spaces have nonzero bundle curvature, and the curvatures of their base surfaces are positive, zero and negative respectively. Any CMC sphere in \(X\) is rotational, see [\textit{U. Abresch} and \textit{H. Rosenberg}, Acta Math. 193, 141--174 (2004; Zbl 1078.53053); Mat. Contemp. 28, 1--28 (2005; Zbl 1118.53036)]. If \(\dim I(X)=3\), then \(X\) is isometric to the Lie group \(\mathrm{Sol}_3\) with a left-invariant metric, which has no rotations. For every \(H>0\), there exists an embedded CMC \(H\)-sphere in \(X\), which is unique up to a left translation, see [\textit{B. Daniel} and \textit{P. Mira}, J. Reine Angew. Math. 685, 1--32 (2013; Zbl 1305.53062); \textit{W. H. Meeks III}, Am. J. Math. 135, 763--775 (2013; Zbl 1277.53057)].
In the present article, the authors prove that if \(X\) is not diffeomorphic to \(\mathbb{R}^3\), then for every \(H\in \mathbb{R}\), there exists a CMC \(H\)-sphere in \(M\) and that if \(X\) is diffeomorphic to \(\mathbb{R}^3\), then a value \(H\in \mathbb{R}\) giving a CMC \(H\)-sphere in \(M\) satisfies \(|H| >\mathrm{Ch} (X)/2\), where \(\mathrm{Ch} (X)\) is the Cheeger constant of \(X\) (Theorem 1.1). See [\textit{W. H. Meeks III}, \textit{P. Mira}, \textit{J. Pérez} and \textit{A. Ros}, Adv. Math. 264, 546--592 (2014; Zbl 1296.53027)] and its references for the Cheeger constant. For \(H\in \mathbb{R}\), if a CMC \(H\)-sphere in \(M\) exists, then it is maximally symmetric; in addition, if \(X\not= \mathbb{S}^2 (\kappa )\times \mathbb{R}\), then a CMC \(H\)-sphere in \(M\) has index \(1\) and nullity \(3\), and it is Alexandrov embedded (Theorem 1.2).
Reviewer: Naoya Ando (Kumamoto)There is no stationary cyclically monotone Poisson matching in 2dhttps://zbmath.org/1527.600162024-02-28T19:32:02.718555Z"Huesmann, Martin"https://zbmath.org/authors/?q=ai:huesmann.martin"Mattesini, Francesco"https://zbmath.org/authors/?q=ai:mattesini.francesco"Otto, Felix"https://zbmath.org/authors/?q=ai:otto.felixThe authors prove that there is no cyclically monotone stationary matching of two independent Poisson processes of unit intensity in dimension \(d = 2\). Therefore, the main result is a theorem of non-existence. The proof combines the harmonic approximation approach with local asymptotics for the two-dimensional matching problem for which the authors give a new self-contained proof using martingale arguments. The interest in this problem is motivated, on the one hand, by work on geometric properties of matching, and on the other hand, by work on optimally coupling random measures. Several extensions and variants of the main theorem are discussed. Previous steps in this direction and the history of the issue are also very carefully discussed.
Reviewer: Yuliya S. Mishura (Kyïv)Periodic measures for a class of SPDEs with regime-switchinghttps://zbmath.org/1527.600472024-02-28T19:32:02.718555Z"Lau, Chun Ho"https://zbmath.org/authors/?q=ai:lau.chun-ho"Sun, Wei"https://zbmath.org/authors/?q=ai:sun.wei|sun.wei.2|sun.wei.1|sun.wei.12|sun.fei|sun.wei.5|sun.wei.7Let \(H\) be a real separable Hilbert space, \(V\) a real reflexive Banach space that is continuously and densely embedded into \(H\), \(V^{\ast }\) the dual space of \(V\), \(\{W(t)\}_{t\geq 0}\) an \(H\)-valued cylindrical Wiener process on a complete filtered probability space \((\Omega ,\mathcal{F},\{\mathcal{F} _{t}\}_{t\geq 0},P)\), \(L(H)\) the space of bounded operators on \(H\), \(L_{2}(H) \) the space of Hilbert-Schmidt operators on \(H\), \(Z\) a real Banach space with norm \(\left\vert \cdot \right\vert \), \(N\) a Poisson random measure on \( (Z,\mathcal{B}(Z))\) with intensity measure \(\nu \), such that \(W\) and \(N\) are independent, and \(\widetilde{N}(dt,dz)=N(dt,dz)-dt\nu (dz)\). The authors consider the stochastic partial differential equation \(dX(t)=A(t,X(t), \Lambda (t))dt+B(t,X(t),\Lambda (t))dW(t)+\int_{\{\left\vert z\right\vert <1\}}G(t,X(t),\Lambda (t),z)\widetilde{N}(dt,dz)+\int_{\{\left\vert z\right\vert \geq 1\}}J(t,X(t),\Lambda (t),z)N(dt,dz)\), with the initial condition \(X(0)=x\in H\). Here, the coefficients \(A:[0,\infty )\times V\times \mathbb{N}\rightarrow V^{\ast }\), \(B:[0,\infty )\times V\times \mathbb{N} \rightarrow L_{2}(H)\) and \(G,J:[0,\infty )\times V\times \mathbb{N}\times Z\rightarrow H\) are measurable. The process \(\Lambda (t)\) has state space \( \mathbb{N}\) and is such that when \(\Delta \rightarrow 0\), \(\mathbb{P} (\Lambda (t+\Delta )=j\mid \Lambda (t)=i,X(t)=x)=q_{ij}(x)\Delta +o(\Delta )\) if \(i\neq j\), and \(\mathbb{P}(\Lambda (t+\Delta )=j\mid \Lambda (t)=i,X(t)=x)=1+q_{ij}(x)\Delta +o(\Delta )\) if \(i=j\), \(\{q_{ij}\}\) being Borel measurable functions on \(H\) such that \(q_{ij}(x)\geq 0\) for any \(x\in H \) and \(i,j\in \mathbb{N}\) with \(i\neq j\) and \(\sum_{j\in \mathbb{N} }q_{ij}(x)=0\) for any \(x\in H\) and \(i\in \mathbb{N}\). The authors assume that \(L=sup_{x\in H,i\in \mathbb{N}}\sum_{j\neq i}q_{ij}(x)<+\infty \). Defining \(\Gamma (x,i,r)=\sum_{j\in \mathbb{N}}(j-i)1_{\Delta _{ij}}(x)(r)\), \((x,i,r)\in H\times N\times \lbrack 0,L]\), \(\Lambda (t)\) can be modeled by \( d\Lambda (t)=\int_{[0,L]}\Gamma (X(t-),\Lambda (t-),r)N_{1}(dt,dr)\), for some Poisson random measure \(N_{1}\) having the Lebesgue measure on \([0,L]\) as its characteristic measure, and the authors assume that \(N_{1}\) is independent of \(W\) and \(N\). Assuming hemicontinuity, local monotonicity, coercivity, and growth hypotheses on \(A\), and growth properties on \(B,H\) and \(\rho \), the authors prove the existence of a unique \(H\times N\)-valued adapted càdlàg process \(\{(X(t),\Lambda (t))\}_{t\in \lbrack 0,T]}\), \(T>0\), such that any \(dt\times \mathbb{P}\)-equivalent class \(\widehat{X}\) of \(X\) belongs to \(L^{\alpha }([0,T];V)\cap L^{2}([0,T];H)\), \(\mathbb{P}\)-a.s., for any \(V\)-valued progressively measurable \(dt\times \mathbb{P}\)-version \( \overline{X}\) of \(X\), \(X(t)=x+\int_{0}^{t}A(s,X(s),\Lambda (s))ds+\int_{0}^{t}B(s,X(s),\Lambda (s))dW(s)+\int_{0}^{t}\int_{\{\left\vert z\right\vert <1\}}G(t,X(t),\Lambda (t),z)\widetilde{N}(dt,dz)+\int_{0}^{t} \int_{\{\left\vert z\right\vert \geq 1\}}J(t,X(t),\Lambda (t),z)N(dt,dz)\), with \(x\in H\), for all \(t\in \lbrack 0,T]\) and \(\mathbb{P}\)-a.s., and \( \Lambda (0)=i\in \mathbb{N}\) and the above stochastic partial differential equation is satisfied.
For the proof, the authors first consider this equation when \(i\in \mathbb{N}\) is fixed, for which they prove an existence result using tools of the literature. They thus obtain a unique \(H\)-valued adapted process \(X^{(i)}(t)\) which satisfies an equality similar to the preceding one. They conclude by induction on the jump points of the stationary point process corresponding to the Poisson random measure \( N_{1}(dt,dr)\). The second main result proves a strong Feller property for the transition semigroup associated with the above stochastic partial differential equation. Let \(P(s,(x,i);t,A)\) be the transition probability function of \(\{(X(t),\Lambda (t))\}_{t\geq 0}\) given by \(P(s,(x,i);t,A)= \mathbb{P}((X(t),\Lambda (t))\in A\mid (X(s),\Lambda (s))=(x,i))\), where \( x\in H\), \(i\in \mathbb{N}\), \(A\in \mathcal{B}(H\times N)\) and \(0\leq s<t<\infty \), and the corresponding time-inhomogeneous transition semigroup be defined by \(P_{s,t}f(x,i)=\int_{H\times \mathbb{N}}f(w)P(s,(x,i);t,dw)\), for every \(f\in B_{b}(H\times \mathbb{N})\). \(\{P_{s,t}\}\) is said to be strong Feller if \(P_{s,t}(B_{b}(H\times \mathbb{N}))\subset C_{b}(H\times \mathbb{N})\), for any \(0\leq s<t<\infty \). \(\{P_{s,t}\}\) is said to be irreducible if \(P(s,(x,i);t,U)>0\) for any \((x,i)\in H\times \mathbb{N}\), non-empty open set \(U\subset H\times \mathbb{N}\) and \(0\leq s<t<\infty \). Assuming further hypotheses on the coefficients involved in the stochastic partial differential equation, the authors prove that the transition semigroup \( \{P_{s,t}\}\) is strong Feller.
The last main result of the paper focuses on a periodic case, with \(l>0\). Now assuming that \(A,B,G,J\) are \(l\)-periodic with respect to \(t\) and satisfy further hypotheses and that the embedding of \(V\) into \(H\) is compact, the authors prove that the stochastic partial differential equation has a unique solution \(\{(X(t),\Lambda (t))\}_{t\geq 0} \), that the transition semigroup \(\{P_{s,t}\}\) of \(\{(X(t),\Lambda (t))\}_{t\geq 0}\) is strong Feller and irreducible, that the hybrid system \( \{(X(t),\Lambda (t))\}_{t\geq 0}\) has a unique \(l\)-periodic measure \(\mu _{0} \), and if \(\mu _{s}(A)=\mathbb{P}_{\mu _{0}}((X(s),\Lambda (s))\in A)\) for \( A\in \mathcal{B}(H\times \mathbb{N})\) and \(s\geq 0\), then, for any \(s\geq 0\) and \(\varphi \in L^{2}(H\times \mathbb{N};\mu _{s})\), \(\lim_{n\rightarrow \infty }\frac{1}{n}\sum_{i=1}^{n}P_{s,s+il}\varphi =\int_{H\times \mathbb{N} }\varphi d\mu _{s}\) in \(L2(H\times \mathbb{N};\mu _{s})\). The proof uses in part the tools already developed in the preceding proofs. The paper ends with an example, the authors considering stochastic fractional porous medium equations.
Reviewer: Alain Brillard (Riedisheim)On the convergence rate of Fletcher-Reeves nonlinear conjugate gradient methods satisfying strong Wolfe conditions: application to parameter identification in problems governed by general dynamicshttps://zbmath.org/1527.650482024-02-28T19:32:02.718555Z"Riahi, Mohamed Kamel"https://zbmath.org/authors/?q=ai:riahi.mohamed-kamel"Qattan, Issam A."https://zbmath.org/authors/?q=ai:qattan.issam-a(no abstract)A first-order Rician denoising and deblurring modelhttps://zbmath.org/1527.650902024-02-28T19:32:02.718555Z"Yang, Wenli"https://zbmath.org/authors/?q=ai:yang.wenli.1"Huang, Zhongyi"https://zbmath.org/authors/?q=ai:huang.zhongyi"Zhu, Wei"https://zbmath.org/authors/?q=ai:zhu.wei.1Summary: In this paper, we propose a first-order variational model for the Rician denoising and deblurring. The model employs a recently developed regularizer that has proven to be effective in image restoration [\textit{W. Zhu}, J. Sci. Comput. 88, No. 2, Paper No. 46, 23 p. (2021; Zbl 1471.94006)]. Due to this regularizer, our model is able to suppress the staircase effect, and more importantly, it helps preserve image contrast, which considerably reduces the blurring effect in restored images. We present the maximum-minimum principle of the model and give a more accurate convex approximation than the conventional one for the fidelity term. Augmented Lagrangian method is utilized to minimize the associated functional. Synthetic and real gray and color images are tested to demonstrate the features of our model.Adaptive space-time finite element methods for parabolic optimal control problemshttps://zbmath.org/1527.650992024-02-28T19:32:02.718555Z"Langer, Ulrich"https://zbmath.org/authors/?q=ai:langer.ulrich"Schafelner, Andreas"https://zbmath.org/authors/?q=ai:schafelner.andreasSummary: We present, analyze, and test locally stabilized space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of space-time tracking parabolic optimal control problems with the standard \(L_2\)-regularization. We derive a priori discretization error estimates in terms of the local mesh-sizes for shape-regular meshes. The adaptive version is driven by local residual error indicators, or, alternatively, by local error indicators derived from a new functional a posteriori error estimator. The latter provides a guaranteed upper bound of the error, but is more costly than the residual error indicators. We perform numerical tests for benchmark examples having different features. In particular, we consider a discontinuous target in form of a first expanding and then contracting ball in 3d that is fixed in the 4d space -- time cylinder.Robust full waveform inversion: a source wavelet manipulation perspectivehttps://zbmath.org/1527.651172024-02-28T19:32:02.718555Z"Bao, Chenglong"https://zbmath.org/authors/?q=ai:bao.chenglong"Qiu, Lingyun"https://zbmath.org/authors/?q=ai:qiu.lingyun"Wang, Rongqian"https://zbmath.org/authors/?q=ai:wang.rongqianSummary: Full waveform inversion (FWI) is a powerful tool for high-resolution subsurface parameter reconstruction. Due to the existence of local minimum traps, the success of the inversion process usually requires a good initial model. Our study primarily focuses on understanding the impact of source wavelets on the landscape of the corresponding optimization problem. We thus introduce a decomposition scheme that divides the inverse problem into two parts. The first step transforms the measured data into data associated with the desired source wavelet. Here, we consider inversions with known and unknown sources to mimic real scenarios. The second subproblem is the conventional FWI, which is much less dependent on an accurate initial model since the previous step improves the misfit landscape. A regularized deconvolution method and a convolutional neural network are employed to solve the source transformation problem. Numerical experiments on the benchmark models demonstrate that our approach improves the gradient's quality in the subsequent FWI and provides a better inversion performance.Adaptive virtual element method for optimal control problem governed by Stokes equationshttps://zbmath.org/1527.651292024-02-28T19:32:02.718555Z"Li, Yanwei"https://zbmath.org/authors/?q=ai:li.yanwei"Wang, Qiming"https://zbmath.org/authors/?q=ai:wang.qiming"Zhou, Zhaojie"https://zbmath.org/authors/?q=ai:zhou.zhaojieSummary: In this paper, adaptive virtual element method (VEM) approximation of optimal control problem governed by Stokes equations with control constraints is discussed. The virtual element discrete scheme of the optimal control problem is constructed by polynomial projections and variational discretization of the control variable. Based on the a posteriori error estimates of VEM for Stokes equations and approximated error equivalence between the solutions of the optimal control problem and the solutions of the state and adjoint equations, we build up upper and lower bounds for the a posteriori error estimates of the optimal control problem. It proves that the a posteriori error indicator is reliable and efficient. The theoretical findings are illustrated by the numerical experiments.Optimised Trotter decompositions for classical and quantum computinghttps://zbmath.org/1527.810342024-02-28T19:32:02.718555Z"Ostmeyer, Johann"https://zbmath.org/authors/?q=ai:ostmeyer.johannSummary: Suzuki-Trotter decompositions of exponential operators like \(\exp(Ht)\) are required in almost every branch of numerical physics. Often the exponent under consideration has to be split into more than two operators \(H = \sum_k A_k\), for instance as local gates on quantum computers. We demonstrate how highly optimised schemes originally derived for exactly two operators \(A_{1, 2}\) can be applied to such generic Suzuki-Trotter decompositions, providing a formal proof of correctness as well as numerical evidence of efficiency. A comprehensive review of existing symmetric decomposition schemes up to order \(n \leqslant 4\) is presented and complemented by a number of novel schemes, including both real and complex coefficients. We derive the theoretically most efficient unitary and non-unitary 4th order decompositions. The list is augmented by several exceptionally efficient schemes of higher order \(n \leqslant 8\). Furthermore we show how Taylor expansions can be used on classical devices to reach machine precision at a computational effort at which state of the art Trotterization schemes do not surpass a relative precision of \(10^{-4}\). Finally, a short and easily understandable summary explains how to choose the optimal decomposition in any given scenario.Optimization, learning algorithms and applications. First international conference, OL2A 2021, Bragança, Portugal, July 19--21, 2021. Revised selected papershttps://zbmath.org/1527.900042024-02-28T19:32:02.718555Z"Lopes, Rui P."https://zbmath.org/authors/?q=ai:lopes.rui-pedroPublisher's description: This book constitutes selected and revised papers presented at the First International Conference on Optimization, Learning Algorithms and Applications, OL2A 2021, held in Bragança, Portugal, in July 2021. Due to the COVID-19 pandemic the conference was held online.
The 39 full papers and 13 short papers were thoroughly reviewed and selected from 134 submissions. They are organized in the topical sections on optimization theory; robotics; measurements with the internet of things; optimization in control systems design; deep learning; data visualization and virtual reality; health informatics; data analysis; trends in engineering education.
The articles of mathematical interest will be reviewed individually.Modern optimization methods for science, engineering and technologyhttps://zbmath.org/1527.900052024-02-28T19:32:02.718555ZPublisher's description: Achieving a better solution or improving the performance of existing system design is an ongoing a process for which scientists, engineers, mathematicians and researchers have been striving for many years. Ever increasingly practical and robust methods have been developed, and every new generation of computers with their increased power and speed allows for the development and wider application of new types of solutions. This book defines the fundamentals, background and theoretical concepts of optimization principles in a comprehensive manner along with their potential applications and implementation strategies. It encompasses linear programming, multivariable methods for risk assessment, nonlinear methods, ant colony optimization, particle swarm optimization, multi-criterion and topology optimization, learning classifier, case studies on six sigma, performance measures and evaluation, multi-objective optimization problems, machine learning approaches, genetic algorithms and quality of service optimizations. The book will be very useful for wide spectrum of target readers including students and researchers in academia and industry.
The articles of this volume will not be indexed individually.Integration methods for aircraft scheduling and trajectory optimization at a busy terminal manoeuvring areahttps://zbmath.org/1527.901072024-02-28T19:32:02.718555Z"Samà, Marcella"https://zbmath.org/authors/?q=ai:sama.marcella"D'Ariano, Andrea"https://zbmath.org/authors/?q=ai:dariano.andrea"Palagachev, Konstantin"https://zbmath.org/authors/?q=ai:palagachev.konstantin"Gerdts, Matthias"https://zbmath.org/authors/?q=ai:gerdts.matthiasSummary: This paper deals with the problem of efficiently scheduling take-off and landing operations at a busy terminal manoeuvring area (TMA). This problem is particularly challenging, since the TMAs are becoming saturated due to the continuous growth of traffic demand and the limited available infrastructure capacity. The mathematical formulation of the problem requires taking into account several features simultaneously: the trajectory of each aircraft should be accurately predicted in each TMA resource, the safety rules between consecutive aircraft need to be modelled with high precision, the aircraft timing and ordering decisions have to be taken in a short time by optimizing performance indicators of practical interest, including the minimization of aircraft delays, travel times and fuel consumption. This work presents alternative approaches to integrate various modelling features and to optimize various performance indicators. The approaches are based on the resolution of mixed-integer linear programs via dedicated solvers. Computational experiments are performed on real-world data from Milano Malpensa in case of multiple delayed aircraft. The results obtained for the proposed approaches show different trade-off solutions when prioritizing different indicators.The polytope of binary sequences with bounded variationhttps://zbmath.org/1527.901342024-02-28T19:32:02.718555Z"Buchheim, Christoph"https://zbmath.org/authors/?q=ai:buchheim.christoph"Hügging, Maja"https://zbmath.org/authors/?q=ai:hugging.majaSummary: We investigate the problem of optimizing a linear objective function over the set of all binary vectors of length \(n\) with bounded variation, where the latter is defined as the number of pairs of consecutive entries with different value. This problem arises naturally in many applications, e.g., in unit commitment problems or when discretizing binary optimal control problems subject to a bounded total variation. We study two variants of the problem. In the first one, the variation of the binary vector is penalized in the objective function, while in the second one, it is bounded by a hard constraint. We show that the first variant is easy to deal with while the second variant turns out to be more complex, but still tractable. For the latter case, we present a complete polyhedral description of the convex hull of feasible solutions by facet-inducing inequalities and devise an exact linear-time separation algorithm. The proof of completeness also yields a new exact primal algorithm with a running time of \(\mathcal{O}(n\log n)\), which is significantly faster than the straightforward dynamic programming approach. Finally, we devise a compact extended formulation.
A preliminary version of this article has been published in [Lect. Notes Comput. Sci. 13526, 64--75 (2022; Zbl 07722405)].Optimality conditions and duality results for a robust bi-level programming problemhttps://zbmath.org/1527.901392024-02-28T19:32:02.718555Z"Saini, Shivani"https://zbmath.org/authors/?q=ai:saini.shivani"Kailey, Navdeep"https://zbmath.org/authors/?q=ai:kailey.navdeep"Ahmad, Izhar"https://zbmath.org/authors/?q=ai:ahmad.izharThe most common approach to develop the optimality conditions and duality theorems of a bi-level programming problem (BLPP) is to transform it to an equivalent single-level mathematical programming problem. This paper deals with a bi-level model whose upper-level constraints include some uncertainty, and the lower-level problem is fully convex. An equivalent single-level mathematical problem is established by using the optimal value reformulation. To deal with uncertainty at the upper-level problem, the concept of robust counterpart optimization is used, and the KKT-type necessary optimality conditions are developed. The authors have extended Abadie Constraint qualifications and introduced an extended non-smooth robust constraint qualification (RCQ). Further as an application, the robust bi-level Mond-Weir dual (MWD) results are developed and the relationship between the solutions to both problems with the help of weak and strong duality theorems are established. Finally, the authors propose some challenging problems for future research.
Reviewer: Samir Kumar Neogy (New Delhi)Characterizing convexity of images for quadratic-linear mappings with applications in nonconvex quadratic optimizationhttps://zbmath.org/1527.901412024-02-28T19:32:02.718555Z"Flores-Bazán, Fabián"https://zbmath.org/authors/?q=ai:flores-bazan.fabian"Opazo, Felipe"https://zbmath.org/authors/?q=ai:opazo.felipeSummary: Various characterizations of convexity for images of a vector mapping where some of its components are quadratic and the remaining ones are linear are established. In a certain sense, one might conclude that convexity of the full image is reduced to the convexity of an image in a lower dimension by deleting the linear components. The latter may be considered as the analogue to the reduction of the number of constraints once the dual is associated. The cases of having one or two quadratic components while the other are linear are particularly analyzed. This allows us to formulate some (geometric) sufficient and necessary conditions for convexity. As a byproduct, a result obtained in [\textit{Y. Xia} et al., Math. Program. 156, No. 1--2 (A), 513--547 (2016; Zbl 1333.90086)] is corrected. Finally, as some applications, we obtain an S-lemma (with equality and on an affine subspace) and a characterization of strong duality in terms of convexity of some image set associated to the minimization problem under consideration.Sufficient conditions for the linear convergence of an algorithm for finding the metric projection of a point onto a convex compact sethttps://zbmath.org/1527.901542024-02-28T19:32:02.718555Z"Balashov, M. V."https://zbmath.org/authors/?q=ai:balashov.maxim-viktorovichSummary: Many problems, for example, problems on the properties of the reachability set of a linear control system, are reduced to finding the projection of zero onto some convex compact subset in a finite-dimensional Euclidean space. This set is given by its support function. In this paper, we discuss some minimum sufficient conditions that must be imposed on a convex compact set so that the gradient projection method for solving the problem of finding the projection of zero onto this set converges with a linear rate. An example is used to illustrate the importance of such conditions.The generalized Bregman distancehttps://zbmath.org/1527.901562024-02-28T19:32:02.718555Z"Burachik, Regina S."https://zbmath.org/authors/?q=ai:burachik.regina-sandra"Dao, Minh N."https://zbmath.org/authors/?q=ai:dao.minh-ngoc"Lindstrom, Scott B."https://zbmath.org/authors/?q=ai:lindstrom.scott-bSummary: Recently, a new kind of distance has been introduced for the graphs of two point-to-set operators, one of which is maximally monotone. When both operators are the subdifferential of a proper lower semicontinuous convex function, this kind of distance specializes under modest assumptions to the classical Bregman distance. We name this new kind of distance the \textit{generalized Bregman distance}, and we shed light on it with examples that utilize the other two most natural representative functions: the Fitzpatrick function and its conjugate. We provide sufficient conditions for convexity, coercivity, and supercoercivity: properties which are essential for implementation in proximal point type algorithms. We establish these results for both the left and right variants of this new kind of distance. We construct examples closely related to the Kullback-Leibler divergence, which was previously considered in the context of Bregman distances and whose importance in information theory is well known. In so doing, we demonstrate how to compute a difficult Fitzpatrick conjugate function, and we discover natural occurrences of the Lambert \(\mathcal{W}\) function, whose importance in optimization is of growing interest.A viscosity iterative method with alternated inertial terms for solving the split feasibility problemhttps://zbmath.org/1527.901612024-02-28T19:32:02.718555Z"Liu, Lulu"https://zbmath.org/authors/?q=ai:liu.lulu"Dong, Qiao-Li"https://zbmath.org/authors/?q=ai:dong.qiaoli"Wang, Shen"https://zbmath.org/authors/?q=ai:wang.shen.1|wang.shen"Rassias, Michael Th."https://zbmath.org/authors/?q=ai:rassias.michael-thSummary: In this paper, we propose a viscosity iterative algorithm with alternated inertial extrapolation step to solve the split feasibility problem, where the self-adaptive stepsize is used. Under appropriate conditions, the proposed algorithm is proved to converge to a solution of the split feasibility problem, which is also the unique solution of a variational inequality problem. Finally, we demonstrate the effectiveness of the algorithm by a numerical example.
For the entire collection see [Zbl 1495.90002].Smoothing fast proximal gradient algorithm for the relaxation of matrix rank regularization problemhttps://zbmath.org/1527.901652024-02-28T19:32:02.718555Z"Zhang, Jie"https://zbmath.org/authors/?q=ai:zhang.jie.1|zhang.jie|zhang.jie.50|zhang.jie.21|zhang.jie.13|zhang.jie.8|zhang.jie.6|zhang.jie.7|zhang.jie.12|zhang.jie.4|zhang.jie.14|zhang.jie.3|zhang.jie.15|zhang.jie.16"Yang, Xinmin"https://zbmath.org/authors/?q=ai:yang.xinmin|yang.xinmin.1Summary: This paper proposes a general inertial smoothing proximal gradient algorithm for solving the Capped-\(\ell_1\) exact continuous relaxation regularization model proposed by \textit{Q. Yu} and \textit{X. Zhang} [Comput. Optim. Appl. 81, No. 2, 519--538 (2022; Zbl 1490.65083)]. The proposed algorithm incorporates different extrapolations into the gradient and proximal steps. It is proved that, under some general parameter constraints, the singular values of any accumulation point of the sequence generated by the proposed algorithm have the common support set, and the zero singular values can be achieved in a finite number of iterations. Furthermore, any accumulation point is a lifted stationary point of the relaxation model. Numerical experiments illustrate the efficiency of the proposed algorithm on synthetic and real data, respectively.Robust strong duality for nonconvex optimization problem under data uncertainty in constrainthttps://zbmath.org/1527.901682024-02-28T19:32:02.718555Z"Chai, Yanfei"https://zbmath.org/authors/?q=ai:chai.yanfeiSummary: This paper deals with the robust strong duality for nonconvex optimization problem with the data uncertainty in constraint. A new weak conjugate function which is abstract convex, is introduced and three kinds of robust dual problems are constructed to the primal optimization problem by employing this weak conjugate function: the robust augmented Lagrange dual, the robust weak Fenchel dual and the robust weak Fenchel-Lagrange dual problem. Characterizations of inequality (1.1) according to robust abstract perturbation weak conjugate duality are established by using the abstract convexity. The results are used to obtain robust strong duality between noncovex uncertain optimization problem and its robust dual problems mentioned above, the optimality conditions for this noncovex uncertain optimization problem are also investigated.A three-operator splitting algorithm with deviations for generalized DC programminghttps://zbmath.org/1527.901722024-02-28T19:32:02.718555Z"Hu, Ziyue"https://zbmath.org/authors/?q=ai:hu.ziyue"Dong, Qiao-Li"https://zbmath.org/authors/?q=ai:dong.qiaoliSummary: In this paper, we introduce a three-operator splitting algorithm with deviations for solving the minimization problem composed of the sum of two convex functions minus a convex and smooth function in a real Hilbert space. The main feature of the proposed method is that two per-iteration deviation vectors provide additional degrees of freedom. We propose one-step and two step inertial three-operator splitting algorithms by selecting the deviations along a momentum direction. A numerical experiment for DC regularized sparse recovery problems shows that the proposed algorithms have better performance than the original three-operator splitting algorithm.Extension of the value function reformulation to multiobjective bilevel optimizationhttps://zbmath.org/1527.901762024-02-28T19:32:02.718555Z"Lafhim, Lahoussine"https://zbmath.org/authors/?q=ai:lafhim.lahoussine"Zemkoho, Alain"https://zbmath.org/authors/?q=ai:zemkoho.alain-bSummary: We consider a multiobjective bilevel optimization problem with vector-valued upper- and lower-level objective functions. Such problems have attracted a lot of interest in recent years. However, so far, scalarization has appeared to be the main approach used to deal with the lower-level problem. Here, we utilize the concept of frontier map that extends the notion of optimal value function to our parametric multiobjective lower-level problem. Based on this, we build a tractable constraint qualification that we use to derive necessary optimality conditions for the problem. Subsequently, we show that our resulting necessary optimality conditions represent a natural extension from standard optimistic bilevel programs with scalar objective functions.Necessary conditions for weak minima and for strict minima of order two in nonsmooth constrained multiobjective optimizationhttps://zbmath.org/1527.901972024-02-28T19:32:02.718555Z"Constantin, Elena"https://zbmath.org/authors/?q=ai:constantin.elenaSummary: In this paper, we give necessary conditions for the existence of a strict local minimum of order two for multiobjective optimization problems with equality and inequality constraints. We suppose that the objective function and the active inequality constraints are only locally Lipschitz. We consider both regular equality constraints and degenerate equality constraints. This article could be considered as a continuation of [E. Constantin, Necessary Conditions for Weak Efficiency for Nonsmooth Degenerate Multiobjective Optimization Problems, J. Global Optim, 75, 111-129, 2019]. We introduce a constraint qualification and a regularity condition, and we show that under each of them, the dual necessary conditions for a weak local minimum of the aforementioned article become of Kuhn-Tucker type.On stationarity for nonsmooth multiobjective problems with vanishing constraintshttps://zbmath.org/1527.902062024-02-28T19:32:02.718555Z"Sadeghieh, Ali"https://zbmath.org/authors/?q=ai:sadeghieh.ali"Kanzi, Nader"https://zbmath.org/authors/?q=ai:kanzi.nader"Caristi, Giuseppe"https://zbmath.org/authors/?q=ai:caristi.giuseppe"Barilla, David"https://zbmath.org/authors/?q=ai:barilla.dSummary: The aim of this paper is to develop first-order necessary and sufficient optimality conditions for nonsmooth multiobjective optimization problems with vanishing constraints. First of all, we introduce some data qualifications for the problem, and derive the comparisons between them. Secondly, based on the mentioned qualifications, we demonstrate some necessary optimality conditions, named strongly stationary conditions, at weakly efficient and efficient solutions of considered problem. Then, we show that the strongly stationary conditions are also sufficient for optimality. Finally, using the tightened problems, we establish other classes of qualifications and stationary conditions.General versions of the Ekeland variational principle: Ekeland points and stop and go dynamicshttps://zbmath.org/1527.902202024-02-28T19:32:02.718555Z"Hai, Le Phuoc"https://zbmath.org/authors/?q=ai:hai.le-phuoc"Khanh, Phan Quoc"https://zbmath.org/authors/?q=ai:phan-quoc-khanh."Soubeyran, Antoine"https://zbmath.org/authors/?q=ai:soubeyran.antoineSummary: We establish general versions of the Ekeland variational principle (EVP), where we include two perturbation bifunctions to discuss and obtain better perturbations for obtaining three improved versions of the principle. Here, unlike the usual studies and applications of the EVP, which aim at exact minimizers via a limiting process, our versions provide good-enough approximate minimizers aiming at applications in particular situations. For the presentation of applications chosen in this paper, the underlying space is a partial quasi-metric one. To prove the aforementioned versions, we need a new proof technique. The novelties of the results are in both theoretical and application aspects. In particular, for applications, using our versions of the EVP together with new concepts of Ekeland points and stop and go dynamics, we study in detail human dynamics in terms of a psychological traveler problem, a typical model in behavioral sciences.Nonsmooth mathematical programs with vanishing constraints in Banach spaceshttps://zbmath.org/1527.902242024-02-28T19:32:02.718555Z"Laha, Vivek"https://zbmath.org/authors/?q=ai:laha.vivek"Singh, Vinay"https://zbmath.org/authors/?q=ai:singh.vinay-pratap|singh.vinay-kumar"Pandey, Yogendra"https://zbmath.org/authors/?q=ai:pandey.yogendra"Mishra, S. K."https://zbmath.org/authors/?q=ai:mishra.shashi-kantSummary: In this chapter, we study the optimization problems with equality, inequality, and vanishing constraints in a Banach space where the objective function and the binding constraints are either differentiable at the optimal solution or Lipschitz near the optimal solution. We derive nonsmooth Karush-Kuhn-Tucker (KKT) type necessary optimality conditions for the above problem where Fréchet (or Gâteaux or Hadamard) derivatives are used for the differentiable functions and the Michel-Penot (M-P) subdifferentials are used for the Lipschitz continuous functions. We also introduce several modifications of some known constraint qualifications like Abadie constraint qualification, Cottle constraint qualification, Slater constraint qualification, Mangasarian-Fromovitz constraint qualification, and linear independence constraint qualification for the above mentioned problem which is called as the nonsmooth mathematical programs with vanishing constraints (NMPVC) in terms of the M-P subdifferentials and establish relationships among them.
For the entire collection see [Zbl 1495.90002].Stability analysis for semi-infinite vector optimization problems under functional perturbationshttps://zbmath.org/1527.902302024-02-28T19:32:02.718555Z"Peng, Zai-Yun"https://zbmath.org/authors/?q=ai:peng.zaiyun"Zhao, Yun-Bin"https://zbmath.org/authors/?q=ai:zhao.yunbin"Yiu, Ka Fai Cedric"https://zbmath.org/authors/?q=ai:yiu.ka-fai-cedric"Zhou, Ya-Cong"https://zbmath.org/authors/?q=ai:zhou.yacongSummary: This paper aims to study the stability of a class of semi-infinite vector optimization problems (SVOP) under functional perturbations. By using an important hypothesis \(\mathbf{H}_h(p_0)\) a necessary and sufficient condition of Hausdorff continuity for weak efficient solution mappings and certain sufficient conditions for Painlevé-Kuratowski convergence of weak efficient solution sets for SVOP are established under the perturbations of both constraint sets and objective functions.Higher-order efficiency conditions for constrained vector equilibrium problemshttps://zbmath.org/1527.902392024-02-28T19:32:02.718555Z"Su, Tran Van"https://zbmath.org/authors/?q=ai:su.tran-van"Luu, Do Van"https://zbmath.org/authors/?q=ai:do-van-luu.Summary: In this paper, we investigate some optimality conditions of higher-order for nonsmooth nonconvex vector equilibrium problems with constraints (or without constraints) in terms of the higher-order upper and lower Studniarski derivatives in Banach spaces. The calculus rule of all the data involved in the problem is taken into account. Using the notion of higher-order upper Studniarski derivative with the class of \(m\)-stable/\(m\)-steady functions, we first provide the higher-order necessary optimality conditions for the local weak efficient solution of vector equilibrium problem without constraints and then we present the higher-order necessary and sufficient optimality conditions for the local strict minimum of order \(m\) to such problem. We second obtain the necessary and sufficient optimality conditions for those efficient solutions of vector equilibrium problem with cone and equality constraints through Lagrange multiplier rules in finite-dimensional spaces. Final, the higher-order necessary and sufficient optimality conditions in terms of the higher-order upper and lower Studniarski derivatives for the local weak efficient solution of vector equilibrium problem with set, cone and equality constraints in Banach spaces are also established. Some examples are proposed to demonstrate our findings.The analytical dynamics of the finite population evolution gameshttps://zbmath.org/1527.910212024-02-28T19:32:02.718555Z"Vardanyan, Edgar"https://zbmath.org/authors/?q=ai:vardanyan.edgar"Saakian, David B."https://zbmath.org/authors/?q=ai:saakian.david-bSummary: We study the dynamics of a finite number of replicators with different strategies in evolutionary games, using the moment closure approximation for the master equation and the Hamilton-Jacobi equation approach. These methods give finite population corrections to the results of the replicator equation. The model under investigation has two strategies \(N\) overall replicators with constant payoff matrix and the Moran process as the update mechanism. Our results are compared with the results of the deterministic replicator equation and the results of numerical stochastic calculations.On the numerical solution of a free end-time homicidal chauffeur gamehttps://zbmath.org/1527.910232024-02-28T19:32:02.718555Z"Calà Campana, Francesca"https://zbmath.org/authors/?q=ai:cala-campana.francesca"De Marchi, Alberto"https://zbmath.org/authors/?q=ai:de-marchi.alberto"Borzì, Alfio"https://zbmath.org/authors/?q=ai:borzi.alfio"Gerdts, Matthias"https://zbmath.org/authors/?q=ai:gerdts.matthiasSummary: A functional formulation of the classical homicidal chauffeur Nash game is presented and a numerical framework for its solution is discussed. This methodology combines a Hamiltonian based scheme with proximal penalty to determine the time horizon where the game takes place with a Lagrangian optimal control approach and relaxation to solve the Nash game at a fixed end-time.A scheme for calculating solvability sets ``up to moment'' in linear differential gameshttps://zbmath.org/1527.910252024-02-28T19:32:02.718555Z"Kamneva, Liudmila"https://zbmath.org/authors/?q=ai:kamneva.liudmilaSummary: A general scheme for calculating a solvability set (backward reachability set) is proposed for a linear conflict controlled system. The backward procedure constructs each subsequent set only based on the set from the previous step. The algorithm is specified for a target set with a convex complement (concave target set). The concavity of solvability sets is justified for a concave target set. The conservation of the concavity singles out a separate class of time-optimal differential games. Note that the convexity is not conserved in the general case of linear time-optimal differential games. On the whole, the article deals with a theoretical approximation of solvability sets in linear time-optimal conflict control problems, the construction of which with structural accuracy (allowing one to distinguish barriers) is usually nontrivial even on a plane. The algorithm is illustrated by numerical calculations for two disturbed dynamics with concave target sets in the plane: the double integrator and the oscillating system.Optimal game theoretic solution of the pursuit-evasion intercept problem using on-policy reinforcement learninghttps://zbmath.org/1527.910262024-02-28T19:32:02.718555Z"Kartal, Yusuf"https://zbmath.org/authors/?q=ai:kartal.yusuf-baris"Subbarao, Kamesh"https://zbmath.org/authors/?q=ai:subbarao.kamesh"Dogan, Atilla"https://zbmath.org/authors/?q=ai:dogan.atilla"Lewis, Frank"https://zbmath.org/authors/?q=ai:lewis.frank-lSummary: This article presents a rigorous formulation for the pursuit-evasion (PE) game when velocity constraints are imposed on agents of the game or players. The game is formulated as an infinite-horizon problem using a non-quadratic functional, then sufficient conditions are derived to prove capture in a finite-time. A novel tracking Hamilton-Jacobi-Isaacs (HJI) equation associated with the non-quadratic value function is employed, which is solved for Nash equilibrium velocity policies for each agent with arbitrary nonlinear dynamics. In contrast to the existing remedies for proof of capture in PE game, the proposed method does not assume players are moving with their maximum velocities and considers the velocity constraints a priori. Attaining the optimal actions requires the solution of HJI equations online and in real-time. We overcome this problem by presenting the on-policy iteration of integral reinforcement learning (IRL) technique. The persistence of excitation for IRL to work is satisfied inherently until capture occurs, at which time the game ends. Furthermore, a nonlinear backstepping control method is proposed to track desired optimal velocity trajectories for players with generalized Newtonian dynamics. Simulation results are provided to show the validity of the proposed methods.
{{\copyright} 2021 John Wiley \& Sons Ltd.}Leveraging online customer reviews in new product development: a differential game approachhttps://zbmath.org/1527.910372024-02-28T19:32:02.718555Z"Liu, Wei"https://zbmath.org/authors/?q=ai:liu.wei.153"Xu, Ke"https://zbmath.org/authors/?q=ai:xu.ke"Chai, Ruirui"https://zbmath.org/authors/?q=ai:chai.ruirui"Fang, Xiang"https://zbmath.org/authors/?q=ai:fang.xiangSummary: Large volumes of online product reviews generated by customers have important strategic values for new product development. We consider a duopoly setting where two manufacturers aim to develop their own new products and services. Applying a differential game framework, we examine how online customer reviews can be leveraged as external knowledge for manufacturers to develop new products. In our base models, we assume that the products supplied by the manufacturers are homogenous. First, we consider a closed innovation setting as a benchmark case in which both manufacturers develop new products by their internal R\&D without leveraging online customer reviews. Second, we propose a model in which one manufacturer leverages online customer reviews as external knowledge, while the other manufacturer only relies on internal R\&D effort. We derive analytical equilibrium solutions to both models. We find that when one manufacturer uses online customer reviews, if the manufacturer's R\&D process becomes more effective in improving its new product performance or reducing its cost, it certainly hurts the other manufacturer, but it may sometimes hurt this particular manufacturer as well. Furthermore, we demonstrate that when the manufacturer utilizes online customer reviews more in R\&D, both manufacturers' profits can either increase or decrease. In an extended model, we relax the product homogeneity assumption and obtain the equilibrium solution analytically. We show that main managerial insights still hold in the extend model.Maximizing gross product for the macroeconomic system with consumption proportional to labor resourceshttps://zbmath.org/1527.911002024-02-28T19:32:02.718555Z"Naumov, V. V."https://zbmath.org/authors/?q=ai:naumov.v-v"Shamaev, I. I."https://zbmath.org/authors/?q=ai:shamaev.i-i"Mestnikov, S. V."https://zbmath.org/authors/?q=ai:mestnikov.semyon-v"Lazarev, N. P."https://zbmath.org/authors/?q=ai:lazarev.nyurgun-petrovich(no abstract)Closed-form solutions of an economic growth model of tourismhttps://zbmath.org/1527.911122024-02-28T19:32:02.718555Z"Irum, Saba"https://zbmath.org/authors/?q=ai:irum.saba"Naeem, Imran"https://zbmath.org/authors/?q=ai:naeem.imran(no abstract)Value functions in a regime switching jump diffusion with delay market modelhttps://zbmath.org/1527.911492024-02-28T19:32:02.718555Z"Llemit, Dennis"https://zbmath.org/authors/?q=ai:llemit.dennis-g"Escaner, Jose Maria L. IV"https://zbmath.org/authors/?q=ai:escaner.jose-maria-l-iv(no abstract)A mean-field game of market-making against strategic tradershttps://zbmath.org/1527.911562024-02-28T19:32:02.718555Z"Baldacci, Bastien"https://zbmath.org/authors/?q=ai:baldacci.bastien"Bergault, Philippe"https://zbmath.org/authors/?q=ai:bergault.philippe"Possamaï, Dylan"https://zbmath.org/authors/?q=ai:possamai.dylanSummary: We design a market-making model à la
\textit{M. Avellaneda} and \textit{S. Stoikov} [Quant. Finance 8, No. 3, 217--224 (2008; Zbl 1152.91024)]
in which the market-takers act strategically, in the sense that they design their trading strategy based on an exogenous trading signal. The market-maker chooses her quotes based on the average market-takers' behavior, modelled through a mean-field interaction. We derive, up to the resolution of a coupled HJB-Fokker-Planck system, the optimal controls of the market-maker and the representative market-taker. This approach is flexible enough to incorporate different behaviors for the market-takers and takes into account the impact of their strategies on the price process.A data-driven optimal control method for endoplasmic reticulum membrane compartmentalization in budding yeast cellshttps://zbmath.org/1527.920032024-02-28T19:32:02.718555Z"Laadhari, Aymen"https://zbmath.org/authors/?q=ai:laadhari.aymen"Barral, Yves"https://zbmath.org/authors/?q=ai:barral.yves"Székely, Gábor"https://zbmath.org/authors/?q=ai:szekely.gabor|szekely.gabor-jSummary: In this paper, we propose a variational data assimilation approach for including data measurements in the simulation of the mobility of fluorescently labeled molecules in the yeast endoplasmic reticulum. The modeling framework aims to provide numerical evidence for compartmentalization in the endoplasmic reticulum. Experimental data are collected and an optimal control problem is formulated as a regularized inverse problem. To our knowledge, this is the first attempt to introduce an optimization formulation constrained by partial differential equations to study the kinetics of fluorescently labeled molecules in budding yeast. We derive the optimality conditions and use an optimize-then-discretize approach. A gradient descent algorithm allows accurate estimation of unknown key parameters in different cellular compartments. The numerical results support the empirical barrier index theory suggesting the presence of a physical diffusion barrier that compartmentalizes the endoplasmic reticulum membrane by limiting the exchange of proteins between the mother and its growing bud. We report several numerical experiments on real data and geometry, with the aim of illustrating the accuracy and efficiency of the method. Furthermore, a relationship between the size ratio of mother and bud compartments and the barrier index ratio is provided.
{{\copyright} 2023 John Wiley \& Sons, Ltd.}Global stability and control strategies of a SIQRS epidemic model with time delayhttps://zbmath.org/1527.920582024-02-28T19:32:02.718555Z"Ma, Yuanyuan"https://zbmath.org/authors/?q=ai:ma.yuanyuan"Cui, Yue"https://zbmath.org/authors/?q=ai:cui.yue"Wang, Min"https://zbmath.org/authors/?q=ai:wang.min.1|wang.min|wang.min.15|wang.min.12|wang.min.2|wang.min.7(no abstract)Modeling SARS-CoV-2 and HBV co-dynamics with optimal controlhttps://zbmath.org/1527.920642024-02-28T19:32:02.718555Z"Omame, Andrew"https://zbmath.org/authors/?q=ai:omame.andrew"Abbas, Mujahid"https://zbmath.org/authors/?q=ai:abbas.mujahidSummary: Clinical reports have shown that chronic hepatitis B virus (HBV) patients co-infected with SARS-CoV-2 have a higher risk of complications with liver disease than patients without SARS-CoV-2. In this work, a co-dynamical model is designed for SARS-CoV-2 and HBV which incorporates incident infection with the dual diseases. Existence of boundary and co-existence endemic equilibria are proved. The occurrence of backward bifurcation, in the absence and presence of incident co-infection, is investigated through the proposed model. It is noted that in the absence of incident co-infection, backward bifurcation is not observed in the model. However, incident co-infection triggers this phenomenon. For a special case of the study, the disease free and endemic equilibria are shown to be globally asymptotically stable. To contain the spread of both infections in case of an endemic situation, the time dependent controls are incorporated in the model. Also, global sensitivity analysis is carried out by using appropriate ranges of the parameter values which helps to assess their level of sensitivity with reference to the reproduction numbers and the infected components of the model. Finally, numerical assessment of the control system using various intervention strategies is performed, and reached at the conclusion that enhanced preventive efforts against incident co-infection could remarkably control the co-circulation of both SARS-CoV-2 and HBV.Controllability and second-order necessary conditions for local infimum trajectories in optimal controlhttps://zbmath.org/1527.930182024-02-28T19:32:02.718555Z"Avakov, E. R."https://zbmath.org/authors/?q=ai:avakov.evgeniy-r"Magaril-Il'yaev, G. G."https://zbmath.org/authors/?q=ai:magaril-ilyaev.gregorij-gA nonlinear control systems is considered on a finite time-interval in the presence of terminal constraints of inequality and equality type under the assumption that the dynamics and the terminal constraints are determined by twice continuously differentiable mappings. The concept for local controllability with respect to a function is defined and a sufficient condition for local controllability is obtained. Next, an optimal control problem of Mayer type is considered and a necessary optimality condition of second-order is proved. Two illustrative examples show the applicability of the obtained results. At last, two approximation lemmas are proved in an Appendix.
Reviewer: Mikhail I. Krastanov (Sofia)Model-free incremental adaptive dynamic programming based approximate robust optimal regulationhttps://zbmath.org/1527.930572024-02-28T19:32:02.718555Z"Li, Cong"https://zbmath.org/authors/?q=ai:li.cong"Wang, Yongchao"https://zbmath.org/authors/?q=ai:wang.yongchao"Liu, Fangzhou"https://zbmath.org/authors/?q=ai:liu.fangzhou"Liu, Qingchen"https://zbmath.org/authors/?q=ai:liu.qingchen"Buss, Martin"https://zbmath.org/authors/?q=ai:buss.martinSummary: This article presents a new formulation for model-free robust optimal regulation of continuous-time nonlinear systems. The proposed reinforcement learning based approach, referred to as incremental adaptive dynamic programming (IADP), utilizes measured input-state data to allow the design of the approximate optimal incremental control strategy, stabilizing the controlled system incrementally under model uncertainties, environmental disturbances, and input saturation. By leveraging the time delay estimation (TDE) technique, we first use sensor data to reduce the requirement of a complete dynamics, where input-state data is adopted to construct an incremental dynamics which reflects the system evolution in an incremental form. Then, the resulting incremental dynamics serves to design the approximate optimal incremental control strategy based on adaptive dynamic programming, which is implemented as a simplified single critic structure to get the approximate solution to the value function of the Hamilton-Jacobi-Bellman equation. Furthermore, for the critic neural network, experience data are used to design an off-policy weight update law with guaranteed weight convergence. Rather importantly, we incorporate a TDE error bound related term into the cost function, whereby the unintentionally introduced TDE error is attenuated during the optimization process. The proofs of system stability and weight convergence are provided. Numerical simulations are conducted to validate the effectiveness and superiority of our proposed IADP, especially regarding the reduced control energy expenditure and the enhanced robustness.
{{\copyright} 2022 The Authors. \textit{International Journal of Robust and Nonlinear Control} published by John Wiley \& Sons Ltd.}A novel domain of attraction based synthesis of inverse optimal controlhttps://zbmath.org/1527.931312024-02-28T19:32:02.718555Z"Prasanna, Parvathy"https://zbmath.org/authors/?q=ai:prasanna.parvathy"Jacob, Jeevamma"https://zbmath.org/authors/?q=ai:jacob.jeevamma"Nandakumar, Mattida P."https://zbmath.org/authors/?q=ai:nandakumar.mattida-pSummary: This article proposes a novel and systematic methodology for the inverse optimal control (IOC) of a class of input affine nonlinear systems. First, the system is represented in the pseudo-linear state-dependent coefficient (SDC) form so as to facilitate the use of matrix algebraic concepts in the design, while preserving the nonlinear system dynamics. Second, the IOC problem which seeks the solution to the Hamilton-Jacobi-Bellman equation is translated to a diagonal stability (D-stability) problem. Therefore, the general notion of D-stability, which deals with the characterization of a matrix \(A\in\Re^{n\times n}\) for the existence of a diagonal matrix \(\Gamma\in\Re^{n\times n}\) such that \(A^T\Gamma+\Gamma A<0\), is redefined with respect to SDC factored matrices in this work. Consequently, the criteria for the existence (\textit{necessary condition}) and determination of a diagonal solution (\textit{sufficient condition}) are formulated in terms of the SDC matrices. It is shown in this work that these criteria can be readily met by the closed-loop system matrix \(\widehat{A}(x)=A(x)-D\), for any arbitrary domain \(S_\delta\) in state-space, by manipulating only the diagonal elements using the diagonal matrix \(D\) constituted by the IOC feedback. In short, the IOC feedback is designed such that the criteria for D-stability by the closed-loop system matrix \(\widehat{A}(x)\) is fulfilled. Hence this novel design methodology is meaningfully named as ``Inverse optimal control via diagonal stabilization'' (IOC-D). The main advantages of IOC-D include: (i) guaranteed estimate of the domain of attraction (DOA), (ii) Immense flexibility of design, and (iii) guaranteed closed form solution which makes the design procedure analytic and computationally efficient. Numerical simulations validate the theoretical developments and the overall efficiency of the proposed approach.
{{\copyright} 2021 John Wiley \& Sons Ltd.}Two-loop reinforcement learning algorithm for finite-horizon optimal control of continuous-time affine nonlinear systemshttps://zbmath.org/1527.931752024-02-28T19:32:02.718555Z"Chen, Zhe"https://zbmath.org/authors/?q=ai:chen.zhe|chen.zhe.1"Xue, Wenqian"https://zbmath.org/authors/?q=ai:xue.wenqian"Li, Ning"https://zbmath.org/authors/?q=ai:li.ning.1"Lewis, Frank L."https://zbmath.org/authors/?q=ai:lewis.frank-lSummary: This article proposes three novel time-varying policy iteration algorithms for finite-horizon optimal control problem of continuous-time affine nonlinear systems. We first propose a model-based time-varying policy iteration algorithm. The method considers time-varying solutions to the Hamiltonian-Jacobi-Bellman equation for finite-horizon optimal control. Based on this algorithm, value function approximation is applied to the Bellman equation by establishing neural networks with time-varying weights. A novel update law for time-varying weights is put forward based on the idea of iterative learning control, which obtains optimal solutions more efficiently compared to previous works. Considering that system models may be unknown in real applications, we propose a partially model-free time-varying policy iteration algorithm that applies integral reinforcement learning to acquiring the time-varying value function. Moreover, analysis of convergence, stability, and optimality is provided for every algorithm. Finally, simulations for different cases are given to verify the convenience and effectiveness of the proposed algorithms.
{{\copyright} 2021 John Wiley \& Sons Ltd.}Adaptive dynamic programming-based event-triggered optimal tracking controlhttps://zbmath.org/1527.932322024-02-28T19:32:02.718555Z"Xue, Shan"https://zbmath.org/authors/?q=ai:xue.shan"Luo, Biao"https://zbmath.org/authors/?q=ai:luo.biao"Liu, Derong"https://zbmath.org/authors/?q=ai:liu.derong"Gao, Ying"https://zbmath.org/authors/?q=ai:gao.ying.1Summary: In this article, an event-triggered constrained optimal tracking control algorithm using integral reinforcement learning (IRL) is developed. First, the constrained optimal tracking control problem is transformed into an optimal regulation problem by employing an augmented system with a discounted value function. Then, IRL is introduced to solve the Hamilton-Jacobi-Bellman equation, where the drift dynamics and reference dynamics are not required. The learning of neural network weights is event-triggered and there is no restriction on the initial control to be admissible. The involvement of event-triggering mechanism alleviates the pressure of data transmission on the network to some extent, which is suitable for control systems with limited computational and communication resources. Moreover, the nonexistence of Zeno behavior and the stability of the impulsive system are proved, respectively. Finally, the application of the algorithm on a mass-spring-damper system verifies its effectiveness.
{{\copyright} 2021 John Wiley \& Sons Ltd.}Adaptive dynamic programming for optimal control of discrete-time nonlinear system with state constraints based on control barrier functionhttps://zbmath.org/1527.932332024-02-28T19:32:02.718555Z"Xu, Jiahui"https://zbmath.org/authors/?q=ai:xu.jiahui"Wang, Jingcheng"https://zbmath.org/authors/?q=ai:wang.jingcheng"Rao, Jun"https://zbmath.org/authors/?q=ai:rao.jun"Zhong, Yanjiu"https://zbmath.org/authors/?q=ai:zhong.yanjiu"Wang, Hongyuan"https://zbmath.org/authors/?q=ai:wang.hongyuanSummary: Adaptive dynamic programming (ADP) methods have demonstrated their efficiency. However, many of the applications for which ADP offers great potential, are also safety-critical and need to meet safety specifications in the presence of physical constraints. In this article, an optimal controller for solving discrete-time nonlinear systems with state constraints is proposed. By introducing the control barrier function into the utility function, the problem with state constraints is transformed into an unconstrained optimal control problem, addressing state constraints which are difficult to handle by traditional ADP methods. The constructed sequence of value function is shown to be monotonically non-increasing and converges to the optimal value. Besides, this article gives the stability proof of the developed algorithm, as well as the conditions for satisfying the state constraints. To implement and approximate the control barrier function based adaptive dynamic programming algorithm, an actor-critic network structure is built. During the training process, two neural networks are used for approximation separately. The performance of the proposed method is validated by testing it on a simulation example.
{{\copyright} 2021 John Wiley \& Sons Ltd.}Suboptimal control for systems with commensurate and distributed delays of neutral typehttps://zbmath.org/1527.932492024-02-28T19:32:02.718555Z"Cuvas, Carlos"https://zbmath.org/authors/?q=ai:cuvas.carlos"Santos-Sánchez, Omar-Jacobo"https://zbmath.org/authors/?q=ai:santos-sanchez.omar-jacobo"Ordaz, Patricio"https://zbmath.org/authors/?q=ai:ordaz.patricio"Rodríguez-Guerrero, Liliam"https://zbmath.org/authors/?q=ai:rodriguez-guerrero.liliamSummary: This article addresses the synthesis of a suboptimal control based on complete type functional for neutral systems with commensurate and distributed delays. The obtained controller is a feedback state control and its gains depend on the time delay system's Lyapunov matrix, which denotes its relevance on the design of controllers for time delay systems. It is proved that in closed loop, the proposed controller improves system response. Despite the synthesized control being suboptimal (optimal in the local sense), it could be a good choice to regulate stable neutral time-delay linear systems without the use of linear matrix inequalities, avoiding the inherent conservatism of such approach. In order to illustrate the result effectiveness, the experimental results are presented by using of an industrial proportional integral derivative (PID) control and LabVIEW software.
{{\copyright} 2021 John Wiley \& Sons Ltd.}Event-triggered optimal tracking control of multiplayer unknown nonlinear systems via adaptive critic designshttps://zbmath.org/1527.933062024-02-28T19:32:02.718555Z"Zhang, Yongwei"https://zbmath.org/authors/?q=ai:zhang.yongwei"Zhao, Bo"https://zbmath.org/authors/?q=ai:zhao.bo"Liu, Derong"https://zbmath.org/authors/?q=ai:liu.derong"Zhang, Shunchao"https://zbmath.org/authors/?q=ai:zhang.shunchaoSummary: In this article, the event-triggered optimal tracking control problem for multiplayer unknown nonlinear systems is investigated by using adaptive critic designs. By constructing a neural network (NN)-based observer with input-output data, the system dynamics of multiplayer unknown nonlinear systems is obtained. Subsequently, the optimal tracking control problem is converted to an optimal regulation problem by establishing a tracking error system. Then, the optimal tracking control policy for each player is derived by solving coupled event-triggered Hamilton-Jacobi (HJ) equation via a critic NN. Meanwhile, a novel weight updating rule is designed by adopting concurrent learning method to relax the persistence of excitation (PE) condition. Moreover, an event-triggering condition is designed by using Lyapunov's direct method to guarantee the uniform ultimate boundedness (UUB) of the closed-loop multiplayer systems. Finally, the effectiveness of the developed method is verified by two different multiplayer nonlinear systems.
{{\copyright} 2021 John Wiley \& Sons Ltd.}Self-learning-based optimal tracking control of an unmanned surface vehicle with pose and velocity constraintshttps://zbmath.org/1527.933202024-02-28T19:32:02.718555Z"Wang, Ning"https://zbmath.org/authors/?q=ai:wang.ning.2"Gao, Ying"https://zbmath.org/authors/?q=ai:gao.ying.1"Liu, Yongjin"https://zbmath.org/authors/?q=ai:liu.yong-jin"Li, Kun"https://zbmath.org/authors/?q=ai:li.kun.7|li.kun.5|li.kun.3|li.kun|li.kun.9|li.kun.2|li.kun.10|li.kun.6|li.kun.8Summary: In this article, subject to both pose and velocity constraints within narrow waters, a self-learning-based optimal tracking control (SLOTC) scheme is innovatively created for an unmanned surface vehicle (USV) by deploying actor-critic reinforcement learning (RL) mechanism and backstepping technique. To be specific, the barrier Lyapunov function (BLF) is devised to uniformly limit the states within a predefined region pertaining to a smoothly feasible reference trajectory. By virtue of a constrained Hamilton-Jacobi-Bellman (HJB) function, an actor-critic control structure under backstepping is established by employing adaptive neural network identifiers which recursively updates actor and critic, simultaneously. Eventually, theoretical analysis proves that the entire SLOTC scheme can render all the states remain in the predefined compact set while tracking errors converge to an arbitrarily small neighborhood of the origin. Simulation results on a prototype USV demonstrate remarkable effectiveness and superiority.
{{\copyright} 2022 John Wiley \& Sons Ltd.}Vertical airborne wind energy farms with high power density per ground area based on multi-aircraft systemshttps://zbmath.org/1527.933222024-02-28T19:32:02.718555Z"De Schutter, Jochem"https://zbmath.org/authors/?q=ai:de-schutter.jochem"Harzer, Jakob"https://zbmath.org/authors/?q=ai:harzer.jakob"Diehl, Moritz"https://zbmath.org/authors/?q=ai:diehl.moritz-mathiasSummary: This paper proposes and simulates vertical airborne wind energy (AWE) farms based on multi-aircraft systems with high power density (PD) per ground area. These farms consist of many independently ground located systems that are flying at the same inclination angle, but with different tether lengths, such that all aircraft fly in a large planar elliptical area that is vertical to the tethers. The individual systems are assigned non-overlapping flight cylinders depending on the wind direction. Detailed calculations that take into account Betz' limit, assuming a cubically averaged wind power density of 7\,m/s, give a potential yearly average PD of 43\,MW/km\(^2\). A conventional wind farm with typical packing density would yield a PD of 2.4\,MW/km\(^2\) in the same wind field. More refined simulations using optimal control result in a more modest PD of 6.5\,MW/km\(^2\) for practically recommended flight trajectories. This PD can already be achieved with small-scale aircraft with a wing span of 5.5\,m. The simulations additionally show that the achievable PD is more than an order of magnitude higher than for a single-aircraft AWE system with the same wing span.Dynamic soaring in wind turbine wakeshttps://zbmath.org/1527.933232024-02-28T19:32:02.718555Z"Harzer, Jakob"https://zbmath.org/authors/?q=ai:harzer.jakob"De Schutter, Jochem"https://zbmath.org/authors/?q=ai:de-schutter.jochem"Diehl, Moritz"https://zbmath.org/authors/?q=ai:diehl.moritz-mathias"Meyers, Johan"https://zbmath.org/authors/?q=ai:meyers.johanSummary: Dynamic soaring for UAVs is a flight technique that enables continuous, powerless periodic flight patterns in the presence of a wind gradient. However, sufficiently large wind gradients are uncommon over land, while at offshore locations the largest wind gradients are located close to the ocean surface, thereby limiting the scope of practical application. An intrinsic feature of wind turbines is that they inherently produce very sharp wind gradients in the near wake. Therefore, in this paper, we propose and investigate periodic stationary dynamic soaring trajectories in the near wake of wind turbines. We additionally consider the potential of dynamic soaring for revitalizing the wind turbine wake. To this end, we apply periodic optimal control based on a simplified model for the glider dynamics and the wind profile in the wake. The cost function maximizes the revitalization of the wake. We compute optimal orbits for a range of different wing spans and different mass-scaling assumptions. The largest glider configuration, with a wingspan of 10\,m and a mass of 222.6\,kg, achieves a wake revitalization of about 0.94\% of the total turbine thrust.Data-driven dynamic relatively optimal controlhttps://zbmath.org/1527.933472024-02-28T19:32:02.718555Z"Pellegrino, Felice A."https://zbmath.org/authors/?q=ai:pellegrino.felice-andrea"Blanchini, Franco"https://zbmath.org/authors/?q=ai:blanchini.franco"Fenu, Gianfranco"https://zbmath.org/authors/?q=ai:fenu.gianfranco"Salvato, Erica"https://zbmath.org/authors/?q=ai:salvato.ericaSummary: We show how the recent works on data-driven open-loop minimum-energy control for linear systems can be exploited to obtain closed-loop control laws in the form of linear dynamic controllers that are relatively optimal. Besides being stabilizing, they achieve the optimal minimum-energy trajectory when the initial condition is the same as the open-loop optimal control problem. The order of the controller is \(N-n\), where \(N\) is the length of the optimal open-loop trajectory, and \(n\) is the order of the system. The same idea can be used for obtaining a relatively optimal controller, entirely based on data, from open-loop trajectories starting from up to \(n\) linearly independent initial conditions.Optimal regulator for a class of nonlinear stochastic systems with random coefficientshttps://zbmath.org/1527.934672024-02-28T19:32:02.718555Z"Algoulity, Mashael"https://zbmath.org/authors/?q=ai:algoulity.mashael"Gashi, Bujar"https://zbmath.org/authors/?q=ai:gashi.bujarSummary: We consider an optimal regulator problem for a class of nonlinear stochastic systems with a \textit{square-root} nonlinearity and \textit{random} coefficients, and using the quadratic-linear criterion. This represents a certain nonlinear generalisation of the stochastic linear-quadratic control problem with random coefficients. The solution if found in an explicit closed-form as an \textit{affine state-feedback} control in terms of a Riccati and linear backward stochastic differential equations. As an application, we give the solution to an optimal investment problem in a market with random coefficients.Linear-quadratic optimal control problems of state delay systems under full and partial informationhttps://zbmath.org/1527.934682024-02-28T19:32:02.718555Z"Chen, Li"https://zbmath.org/authors/?q=ai:chen.li.69|chen.li.62|chen.li.70|chen.li.71|chen.li.58|chen.li.61|chen.li.60|chen.li.57|chen.li.19|chen.li.20|chen.li.10|chen.li.6|chen.li.5|chen.li.1|chen.li.14|chen.li.16|chen.li.9|chen.li"Zhang, Yi"https://zbmath.org/authors/?q=ai:yi.zhang|zhang.yi|zhang.yi.18|zhang.yi.3|zhang.yi.14|zhang.yi.34|zhang.yi.46|zhang.yi.2|zhang.yi.8|zhang.yi.1|zhang.yi.10|zhang.yi.12|zhang.yi.5|zhang.yi.17|zhang.yi.6|zhang.yi.4The authors of this paper study stochastic linear-quadratic optimal control problems of state delay systems under full and partial information. When the full information is available, they study both discrete and distributed delays in the control system. They first characterize the optimal control by Hamiltonian system which consists of distributed delayed SDE and distributed anticipated BSDE. Then they obtain an optimal feedback control by the dynamic programming method. Then they study partially observed problem with state delay and get the optimal control by backward separation method and optimal filtering with delay. Unlike the system with full information, when only partial information is available, they find the optimal feedback control in a particular case.
Reviewer: Qi Lu (Chengdu)Nonlinear-nonquadratic optimal and inverse optimal control for discrete-time stochastic dynamical systemshttps://zbmath.org/1527.934692024-02-28T19:32:02.718555Z"Lanchares, Manuel"https://zbmath.org/authors/?q=ai:lanchares.manuel"Haddad, Wassim M."https://zbmath.org/authors/?q=ai:haddad.wassim-mSummary: In this article, we investigate the role of Lyapunov functions in evaluating nonlinear-nonquadratic cost functionals for Itô-type nonlinear stochastic difference equations. Specifically, it is shown that the cost functional can be evaluated in closed-form as long as the cost functional is related in a specific way to an underlying Lyapunov function that guarantees asymptotic stability in probability. This result is then used to analyze discrete-time linear as well as nonlinear stochastic dynamical systems with polynomial and multilinear cost functionals. Furthermore, a stochastic optimal control framework is developed by exploiting connections between stochastic Lyapunov theory and stochastic Bellman theory. In particular, we show that asymptotic and geometric stability in probability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that can clearly be seen to be the solution to the steady state form of the stochastic Bellman equation, and hence, guaranteeing both stochastic stability and optimality.
{{\copyright} 2021 John Wiley \& Sons Ltd.}