Recent zbMATH articles in MSC 49https://zbmath.org/atom/cc/492021-11-25T18:46:10.358925ZWerkzeug``Controlled'' versions of the Collatz-Wielandt and Donsker-Varadhan formulaehttps://zbmath.org/1472.150092021-11-25T18:46:10.358925Z"Arapostathis, Aristotle"https://zbmath.org/authors/?q=ai:arapostathis.aristotle"Borkar, Vivek S."https://zbmath.org/authors/?q=ai:borkar.vivek-shripadSummary: This is an overview of the work of the authors and their collaborators on the characterization of risk-sensitive costs and rewards in terms of an abstract Collatz-Wielandt formula and in case of rewards, also a controlled version of the Donsker-Varadhan formula. For the finite state and action case, this leads to useful linear and dynamic programming formulations for the reward maximization problem in the reducible case.
For the entire collection see [Zbl 1468.60003].Nearest \(\Omega \)-stable matrix via Riemannian optimizationhttps://zbmath.org/1472.150302021-11-25T18:46:10.358925Z"Noferini, Vanni"https://zbmath.org/authors/?q=ai:noferini.vanni"Poloni, Federico"https://zbmath.org/authors/?q=ai:poloni.federico-gSummary: We study the problem of finding the nearest \(\varOmega \)-stable matrix to a certain matrix \(A\), i.e., the nearest matrix with all its eigenvalues in a prescribed closed set \(\varOmega \). Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.Potential theory on minimal hypersurfaces. I: Singularities as Martin boundarieshttps://zbmath.org/1472.300272021-11-25T18:46:10.358925Z"Lohkamp, Joachim"https://zbmath.org/authors/?q=ai:lohkamp.joachimThe author develops a detailed potential theory on (almost) minimizing hypersurfaces applicable to large classes of linear elliptic second-order operators.
Let \(H\) be an (almost) minimizing hypersurface containing the singularity set \(\Sigma \subset H\). By \(\mathcal S\)-uniformity, we can regard \(H \setminus \Sigma\) as a generalized convex set and \(\Sigma\) as its boundary. Then the author proves a generalized boundary Harnack inequality, and use it to deduce other interesing results concerning Martin theory, the Dirichlet problem and Hardy inequalities.Evolution differential inclusions associated to primal lower regular functionshttps://zbmath.org/1472.340342021-11-25T18:46:10.358925Z"Kecis, Ilyas"https://zbmath.org/authors/?q=ai:kecis.ilyas"Thibault, Lionel"https://zbmath.org/authors/?q=ai:thibault.lionelSummary: In this paper, we discuss the evolution inclusion governed by the subdifferential of a function \(f\) with a perturbation \(g\) depending on the time and the state. We prove the existence and uniqueness of local and global solution, assuming that \(f\) is primal lower regular and \(g\) satisfies some standard conditions. We give also many properties of the solution and study some particular cases of the inclusion, for example, the case when \(g\) depends only on the state and \(u_0\) belongs to the domain of the subdifferential of \(f\).Optimal tax policy of a one-predator-two-prey system with a marine protected areahttps://zbmath.org/1472.340892021-11-25T18:46:10.358925Z"Huang, Lirong"https://zbmath.org/authors/?q=ai:huang.lirong"Cai, Donghan"https://zbmath.org/authors/?q=ai:cai.donghan"Liu, Weiyi"https://zbmath.org/authors/?q=ai:liu.weiyiSummary: This paper is devoted to handle a dynamic one-predator-two-prey model. In order to protect fish population from over exploitation, we assume that marine protected area (MPA) is established and the fisherman only harvest the prey in the unreserved area, the predator consumes the prey in both the MPA and the unreserved area. And a tax is imposed in the process of harvesting. To begin with, boundeness of the system is discussed. Following this, we studied the existence of the possible equilibrium along with their local stability for both of the unexploited system (8) and exploited system (5). After that, we analyzed the global stability of the positive equilibrium of the exploited system and how the tax \(\tau\) affects the positive equilibrium. Then, the optimal tax policy is obtained by using the Pontryagin's maximum principle. Finally, some numerical simulations are given to support the analytical findings.He-Laplace variational iteration method for solving the nonlinear equations arising in chemical kinetics and population dynamicshttps://zbmath.org/1472.350092021-11-25T18:46:10.358925Z"Nadeem, Muhammad"https://zbmath.org/authors/?q=ai:nadeem.muhammad|nadeem.muhammad-faisal"He, Ji-Huan"https://zbmath.org/authors/?q=ai:he.ji-huan|he.jihuanSummary: In this article, we suggest an alternative approach for the study of some partial differential equations (PDEs) arising in physical phenomena such as chemical kinetics and population dynamics. He-Laplace variational iteration method (He-LVIM) is a simple but effective way where the variational iteration method (VIM) is coupled with Laplace transforms method. He's polynomials are obtained by the homotopy perturbation method (HPM) to deal with the nonlinear terms. Fisher equation, the generalized Fisher equation and the nonlinear diffusion equation of the Fisher type are suggested to demonstrate the accuracy and stability of this method.Homogenization of random convolution energieshttps://zbmath.org/1472.350272021-11-25T18:46:10.358925Z"Braides, Andrea"https://zbmath.org/authors/?q=ai:braides.andrea"Piatnitski, Andrey"https://zbmath.org/authors/?q=ai:piatnitski.andrey-lSummary: We prove a homogenization theorem for a class of quadratic convolution energies with random coefficients. Under suitably stated hypotheses of ergodicity and stationarity, we prove that the \(\Gamma\)-limit of such energy is almost surely a deterministic quadratic Dirichlet-type integral functional, whose integrand can be characterized through an asymptotic formula. The proof of this characterization relies on results on the asymptotic behaviour of subadditive processes. The proof of the limit theorem uses a blow-up technique common for local energies, which can be extended to this `asymptotically local' case. As a particular application, we derive a homogenization theorem on random perforated domains.Remarks on parabolic De Giorgi classeshttps://zbmath.org/1472.350772021-11-25T18:46:10.358925Z"Liao, Naian"https://zbmath.org/authors/?q=ai:liao.naianIn ths very interesting paper, the author makes several remarks concerning properties of functions in parabolic De Giorgi classes of order p. This effort is motivated by the fact that There are new perspectives including a novel mechanism of propagating positivity in measure, the reservation of membership under convex composition, and a logarithmic type estimate. Based on them, the author is able to give new proofs of known properties. In particular, local boundedness and local Hölder continuity of these functions via Moser's ideas, thus avoiding De Giorgi's heavy machinery, are proved. The author also takes the opportunity to give a transparent proof of a weak Harnack inequality for nonnegative members of some super-class of De Giorgi, surprisingly without any covering argument.Local singular characteristics on \(\mathbb{R}^2\)https://zbmath.org/1472.350952021-11-25T18:46:10.358925Z"Cannarsa, Piermarco"https://zbmath.org/authors/?q=ai:cannarsa.piermarco"Cheng, Wei"https://zbmath.org/authors/?q=ai:cheng.weiSummary: The singular set of a viscosity solution to a Hamilton-Jacobi equation is known to propagate, from any noncritical singular point, along singular characteristics which are curves satisfying certain differential inclusions. In the literature, different notions of singular characteristics were introduced. However, a general uniqueness criterion for singular characteristics, not restricted to mechanical systems or problems in one space dimension, is missing at the moment. In this paper, we prove that, for a Tonelli Hamiltonian on \(\mathbb{R}^2\), two different notions of singular characteristics coincide up to a bi-Lipschitz reparameterization. As a significant consequence, we obtain a uniqueness result for the class of singular characteristics that was introduced by \textit{K. Khanin} and \textit{A. Sobolevski} in the paper [Arch. Ration. Mech. Anal. 219, No. 2, 861--885 (2016; Zbl 1333.35201)].Sensitivity of the second order homogenized elasticity tensor to topological microstructural changeshttps://zbmath.org/1472.351452021-11-25T18:46:10.358925Z"Calisti, V."https://zbmath.org/authors/?q=ai:calisti.v"Lebée, A."https://zbmath.org/authors/?q=ai:lebee.arthur"Novotny, A. A."https://zbmath.org/authors/?q=ai:novotny.antonio-andre"Sokolowski, J."https://zbmath.org/authors/?q=ai:sokolowski.janSummary: The multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.Positive solutions of the prescribed mean curvature equation with exponential critical growthhttps://zbmath.org/1472.351662021-11-25T18:46:10.358925Z"Figueiredo, Giovany M."https://zbmath.org/authors/?q=ai:figueiredo.giovany-malcher"Rădulescu, Vicenţiu D."https://zbmath.org/authors/?q=ai:radulescu.vicentiu-dGiven a bounded domain \(\Omega \subset \mathbb{R}^2\) with smooth boundary \(\partial\Omega\), the authors study the prescribed mean curvature equation given by
\begin{align*}
-\text{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=f(u) \quad\text{in }\Omega, \quad u=0 \quad\text{on }\partial\Omega,
\end{align*}
where \(f\colon\mathbb{R}\to\mathbb{R}\) is a superlinear continuous function with critical exponential growth. Based on an auxiliary problem along with the Nehari manifold by using Moser's iteration method and Stampacchia's estimates, the existence of a positive solution of the problem above is shown.Computing multiple solutions of topology optimization problemshttps://zbmath.org/1472.353082021-11-25T18:46:10.358925Z"Papadopoulos, Ioannis P. A."https://zbmath.org/authors/?q=ai:papadopoulos.ioannis-p-a"Farrell, Patrick E."https://zbmath.org/authors/?q=ai:farrell.patrick-e"Surowiec, Thomas M."https://zbmath.org/authors/?q=ai:surowiec.thomasExistence and limiting behavior of min-max solutions of the Ginzburg-Landau equations on compact manifoldshttps://zbmath.org/1472.353682021-11-25T18:46:10.358925Z"Stern, Daniel"https://zbmath.org/authors/?q=ai:stern.danielSummary: We use a natural two-parameter min-max construction to produce critical points of the Ginzburg-Landau functionals on a compact Riemannian manifold of dimension \(\geq 2\). We investigate the limiting behavior of these critical points as \(\varepsilon \to 0\), and show in particular that some of the energy concentrates on a nontrivial stationary, rectifiable \((n-2)\)-varifold as \(\epsilon \to 0\), suggesting connections to the min-max construction of minimal \((n-2)\)-submanifolds.Rate of convergence for periodic homogenization of convex Hamilton-Jacobi equations in one dimensionhttps://zbmath.org/1472.353732021-11-25T18:46:10.358925Z"Tu, Son N. T."https://zbmath.org/authors/?q=ai:tu.son-n-tSummary: Let \(u^\varepsilon\) and \(u\) be viscosity solutions of the oscillatory Hamilton-Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of convergence \(\mathcal{O}(\varepsilon)\) of \(u^{\varepsilon}\to u\) as \(\varepsilon\to 0^+\) for a large class of convex Hamiltonians \(H(x,y,p)\) in one dimension. This class includes the Hamiltonians from classical mechanics with separable potential. The proof makes use of optimal control theory and a quantitative version of the ergodic theorem for periodic functions in dimension \(n=1\).Topological sensitivity analysis for a three-dimensional parabolic type problemhttps://zbmath.org/1472.353782021-11-25T18:46:10.358925Z"Ghezaiel, Emna"https://zbmath.org/authors/?q=ai:ghezaiel.emna"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: This work focuses on the topological sensitivity analysis of a three-dimensional parabolic type problem. The considered application model is described by the heat equation. We derive a new topological asymptotic expansion valid for various shape functions and geometric perturbations of arbitrary form. The used approach is based on a rigorous mathematical framework describing and analyzing the asymptotic behavior of the perturbed temperature field.The stability principle and global weak solutions of the free surface semi-geostrophic equations in geostrophic coordinateshttps://zbmath.org/1472.353912021-11-25T18:46:10.358925Z"Cullen, M. J. P."https://zbmath.org/authors/?q=ai:cullen.michael-john-priestley"Kuna, T."https://zbmath.org/authors/?q=ai:kuna.tobias"Pelloni, B."https://zbmath.org/authors/?q=ai:pelloni.beatrice"Wilkinson, M."https://zbmath.org/authors/?q=ai:wilkinson.malcolm-h|wilkinson.mark-a|wilkinson.mark-d|wilkinson.michael-h-fSummary: The semi-geostrophic equations are used widely in the modelling of large-scale atmospheric flows. In this note, we prove the global existence of weak solutions of the incompressible semi-geostrophic equations, in geostrophic coordinates, in a three-dimensional domain with a free upper boundary. The proof, based on an energy minimization argument originally inspired by the Stability Principle as studied by Cullen, Purser and others, uses optimal transport techniques as well as the analysis of Hamiltonian ODEs in spaces of probability measures as studied by Ambrosio and Gangbo. We also give a general formulation of the Stability Principle in a rigorous mathematical framework.Fine metrizable convex relaxations of parabolic optimal control problemshttps://zbmath.org/1472.354152021-11-25T18:46:10.358925Z"Roubíček, Tomáš"https://zbmath.org/authors/?q=ai:roubicek.tomasThe paper deals with fine metrizable convex relaxations of parabolic optimal control problems. In particular, a compromising convex compactification is devised. The basic idea consists in combining classical techniques for Young measures with Choquet theory. Therefore, the proposed approach works under classical \(\sigma\)-additive measures and standard sequences. At the same time, it allows for dealing with a wider class of nonlinearities than only affine. The controls \(u\) are valued in the set \(S_p\) of the form \[ S_p = \{ u \in L^p(\Omega;\mathbb{R}^m): u(x) \in B \; \text{for a.a.} \; x \in \Omega\}, \] with \(\Omega \subset \mathbb{R}^d\), \(d \in \mathbb{N}\), \(1 \leq p < +\infty\) and \(B \subset \mathbb{R}^m\) bounded and closed. In addition, some generalization to unbounded domain \(B\) by considering a general \(S_p\) bounded in \(L^p(\Omega;\mathbb{R}^m)\), with \(1 \leq p < \infty\) fixed but not necessarily bounded in \(L^\infty(\Omega;\mathbb{R}^m)\), is also discussed. Finally, an application to optimal control of a system of semilinear parabolic differential equations is presented for the reader convenience, together with other relaxation strategies as well as more general nonlinearities, showing that the finds reported in the paper are useful for practical applications.Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphshttps://zbmath.org/1472.354182021-11-25T18:46:10.358925Z"Hofmann, Matthias"https://zbmath.org/authors/?q=ai:hofmann.matthias"Kennedy, James B."https://zbmath.org/authors/?q=ai:kennedy.james-bSummary: We prove interlacing inequalities between spectral minimal energies of metric graphs built on Dirichlet and standard Laplacian eigenvalues, as recently introduced in [the second author et al., Calc. Var. Partial Differ. Equ. 60, No. 2, Paper No. 61, 63 p. (2021; Zbl 1462.35222)]. These inequalities, which involve the first Betti number and the number of degree one vertices of the graph, recall both interlacing and other inequalities for the Laplacian eigenvalues of the whole graph, as well as estimates on the difference between the number of nodal and Neumann domains of the whole graph eigenfunctions. To this end we study carefully the principle of \textit{cutting} a graph, in particular quantifying the size of a cut as a perturbation of the original graph via the notion of its \textit{rank}. As a corollary we obtain an inequality between these energies and the actual Dirichlet and standard Laplacian eigenvalues, valid for all compact graphs, which complements a version for tree graphs of Friedlander's inequalities between Dirichlet and Neumann eigenvalues of a domain. In some cases this results in better Laplacian eigenvalue estimates than those obtained previously via more direct methods.A variational inequality based stochastic approximation for inverse problems In stochastic partial differential equationshttps://zbmath.org/1472.354512021-11-25T18:46:10.358925Z"Hawks, Rachel"https://zbmath.org/authors/?q=ai:hawks.rachel"Jadamba, Baasansuren"https://zbmath.org/authors/?q=ai:jadamba.baasansuren"Khan, Akhtar A."https://zbmath.org/authors/?q=ai:khan.akhtar-ali"Sama, Miguel"https://zbmath.org/authors/?q=ai:sama.miguel"Yang, Yidan"https://zbmath.org/authors/?q=ai:yang.yidanSummary: The primary objective of this work is to study the inverse problem of identifying a parameter in partial differential equations with random data. We explore the nonlinear inverse problem in a variational inequality framework. We propose a projected-gradient-type stochastic approximation scheme for general variational inequalities and give a complete convergence analysis under weaker conditions on the random noise than those commonly imposed in the available literature. The proposed iterative scheme is tested on the inverse problem of parameter identification. We provide a derivative characterization of the solution map, which is used in computing the derivative of the objective map. By employing a finite element based discretization scheme, we derive the discrete formulas necessary to test the developed stochastic approximation scheme. Preliminary numerical results show the efficacy of the developed framework.
For the entire collection see [Zbl 1470.49002].On the existence and essential components of solution sets for systems of generalized quasi-variational relation problemshttps://zbmath.org/1472.470542021-11-25T18:46:10.358925Z"Nguyen Van Hung"https://zbmath.org/authors/?q=ai:nguyen-van-hung."Phan Thanh Kieu"https://zbmath.org/authors/?q=ai:phan-thanh-kieu.Summary: In this paper, we study the existence of a solution for a system of quasi-variational relation problems (in short, (SQVR)). Moreover, we discuss the existence of essentially connected components of the solution set for (SQVR). Then the obtained results are applied to systems of quasi-variational inclusions and to systems of weak vector quasi-equilibrium problems. The results presented in the paper improve and extend many results from the literature. Some examples are given to illustrate our results.On solutions of variational inequality problems via iterative methodshttps://zbmath.org/1472.470582021-11-25T18:46:10.358925Z"Alghamdi, Mohammed Ali"https://zbmath.org/authors/?q=ai:alghamdi.mohammed-ali"Shahzad, Naseer"https://zbmath.org/authors/?q=ai:shahzad.naseer"Zegeye, Habtu"https://zbmath.org/authors/?q=ai:zegeye.habtuSummary: We investigate an algorithm for a common point of fixed points of a finite family of Lipschitz pseudocontractive mappings and solutions of a finite family of \(\gamma\)-inverse strongly accretive mappings. Our theorems improve and unify most of the results that have been proved in this direction for this important class of nonlinear mappings.Iterative scheme for solving a class of set-valued variational inequality problemshttps://zbmath.org/1472.470612021-11-25T18:46:10.358925Z"Al-Shemas, Eman"https://zbmath.org/authors/?q=ai:al-shemas.eman-hamad(no abstract)Hybrid extragradient method with regularization for convex minimization, generalized mixed equilibrium, variational inequality and fixed point problemshttps://zbmath.org/1472.470632021-11-25T18:46:10.358925Z"Ceng, Lu-Chuan"https://zbmath.org/authors/?q=ai:ceng.lu-chuan"Ho, Juei-Ling"https://zbmath.org/authors/?q=ai:ho.juei-lingSummary: We introduce two iterative algorithms by the hybrid extragradient method with regularization for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings and the set of fixed points of an asymptotically \(\kappa\)-strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under mild conditions.Iterative schemes for convex minimization problems with constraintshttps://zbmath.org/1472.470652021-11-25T18:46:10.358925Z"Ceng, Lu-Chuan"https://zbmath.org/authors/?q=ai:ceng.lu-chuan"Liao, Cheng-Wen"https://zbmath.org/authors/?q=ai:liao.cheng-wen"Pang, Chin-Tzong"https://zbmath.org/authors/?q=ai:pang.chin-tzong"Wen, Ching-Feng"https://zbmath.org/authors/?q=ai:wen.chingfengSummary: We first introduce and analyze one implicit iterative algorithm for finding a solution of the minimization problem for a convex and continuously Fréchet differentiable functional, with constraints of several problems: the generalized mixed equilibrium problem, the system of generalized equilibrium problems, and finitely many variational inclusions in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another implicit iterative algorithm for finding a fixed point of infinitely many nonexpansive mappings with the same constraints, and derive its strong convergence under mild assumptions.Strong convergence of iterative algorithms for the split equality problemhttps://zbmath.org/1472.470812021-11-25T18:46:10.358925Z"Shi, Luo Yi"https://zbmath.org/authors/?q=ai:shi.luoyi"Chen, Rudong"https://zbmath.org/authors/?q=ai:chen.rudong"Wu, Yujing"https://zbmath.org/authors/?q=ai:wu.yujingSummary: Let \(H_1, H_2, H_3\) be real Hilbert spaces, \(C \subseteq H_1\), \(Q \subseteq H_2\) be two nonempty closed convex sets, and let \(A : H_1 \to H_3\), \(B : H_2 \to H_3\) be two bounded linear operators. The split equality problem (SEP) is finding \(x \in C\), \(y \in Q\) such that \(Ax = By\). Recently, Moudafi has presented the ACQA algorithm and the RACQA algorithm to solve SEP [\textit{A. Moudafi}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 79, 117--121 (2013; Zbl 1256.49044)]. However, the two algorithms are weakly convergent. It is therefore the aim of this paper to construct new algorithms for SEP so that strong convergence is guaranteed. Firstly, we define the concept of the minimal norm solution of SEP. Using Tychonov regularization, we introduce two methods to get such a minimal norm solution. And then, we introduce two algorithms which are viewed as modifications of Moudafi's ACQA, RACQA algorithms and KM-CQ algorithm, respectively, and converge strongly to a solution of SEP. More importantly, the modifications of Moudafi's ACQA, RACQA algorithms converge strongly to the minimal norm solution of SEP. At last, we introduce some other algorithms which converge strongly to a solution of SEP.Projection methods for a system of nonlinear mixed variational inequalities in Banach spaceshttps://zbmath.org/1472.470862021-11-25T18:46:10.358925Z"Wang, Zhong-Bao"https://zbmath.org/authors/?q=ai:wang.zhongbao"Tang, Guo-Ji"https://zbmath.org/authors/?q=ai:tang.guoji"Zhang, Hong-Ling"https://zbmath.org/authors/?q=ai:zhang.honglingSummary: The existence and uniqueness of solution for a system of nonlinear mixed variational inequality in Banach spaces is given firstly. A Mann iterative sequences with errors for this system of nonlinear mixed variational inequalities in Banach spaces is studied, by using the generalized \(f\)-projection operator \(\pi_K^f\). Our results extend the main results in [\textit{R. U. Verma}, Appl. Math. Lett. 18, No. 11, 1286--1292 (2005; Zbl 1099.47054)]; Comput. Math. Appl. 41, No. 7--8, 1025--1031 (2001; Zbl 0995.47042)] from Hilbert spaces to Banach spaces.An invitation to optimal transport, Wasserstein distances, and gradient flowshttps://zbmath.org/1472.490012021-11-25T18:46:10.358925Z"Figalli, Alessio"https://zbmath.org/authors/?q=ai:figalli.alessio"Glaudo, Federico"https://zbmath.org/authors/?q=ai:glaudo.federicoThis graduate text offers a relatively self-contained introduction to the optimal transport theory. It consists of five chapters and two appendices.
Chapter 1 gives a brief review of the optimal transport theory, recalls certain of basics of measure theory and Riemannian geometry, and shows three typical examples of the transport maps in connection to the classical isoperimetry.
Chapter 2 presents the so-called core of the optimal transport theory: the solution to Kantorovich's problem for general costs; the duality theory; the solution to Monge's problem for suitable costs.
Chapter 3 utilizes the \([1,\infty)\ni p\)-Wasserstein distances to handle an essential relationship among the optimal transport theory, gradient flows in the Hilbert spaces, and partial differential equations.
Chapter 4 shows a differential viewpoint of the optimal transport theory via studying Benamou-Brenier's and Otto's formulas based on the probability measures.
Chapter 5 suggests several applied topics of the optimal transport theory.
Appendix A includes a set of some interesting exercises and their solutions.
Appendix B outlines a proof of the disintegration theorem.An invitation to the study of a uniqueness problemhttps://zbmath.org/1472.490022021-11-25T18:46:10.358925Z"Ricceri, Biagio"https://zbmath.org/authors/?q=ai:ricceri.biagioSummary: In this very short chapter, we provide a strong motivation for the study of the following problem: given a real normed space \(E\), a closed, convex, unbounded set \(X \subseteq E\), and a function \(f : X \rightarrow X\), find suitable conditions under which, for each \(y \in X\), the function
\[ x\to \|x-f(x)\|-\|y-f(x)\|\]
has at most one global minimum in \(X\).
For the entire collection see [Zbl 1470.49002].Investigation of the problem of optimal control by a system ODE of block structure with blocks connected only by boundary conditionshttps://zbmath.org/1472.490032021-11-25T18:46:10.358925Z"Aida-Zade, Kamil"https://zbmath.org/authors/?q=ai:aida-zade.kamil-r"Ashrafova, Yegana"https://zbmath.org/authors/?q=ai:ashrafova.yegana-ramizSummary: The paper investigates the problem of optimal control by the systems of ODE of block-structure with unseparated boundary conditions between blocks. The considered complex object consists of blocks. The state of each of them is described by the system of ordinary differential equations. The blocks are interconnected in an arbitrary order only by the initial and/or final (boundary) state values. The necessary optimality conditions for the considered optimal control problem are obtained.
Note that, the obtained adjoint problem has the same specifics as the direct problem. For the numerical solution of optimal control problem, it is proposed to apply first-order optimization methods using the formulas for the functional gradient that participate in the necessary optimality conditions. To solve the direct and adjoint initial boundary-value problems of a block structure and with unseparated nonlocal boundary conditions with sparse (arbitrary filled) matrix, special schemes of the sweep method are proposed that take into account the specifics of the systems of differential equations and boundary conditions that allow the transfer of boundary conditions for each block separately.
For the entire collection see [Zbl 1460.90005].On the existence of solutions to one-dimensional fourth-order equationshttps://zbmath.org/1472.490042021-11-25T18:46:10.358925Z"Shokooh, S."https://zbmath.org/authors/?q=ai:shokooh.saeid"Afrouzi, G. A."https://zbmath.org/authors/?q=ai:afrouzi.ghasem-alizadeh"Hadjian, A."https://zbmath.org/authors/?q=ai:hadjian.arminIn this paper, the authors consider the following fourth-order boundary-value problem: \begin{align*} & u^{(4)}h(x,u')-u''=[\lambda f(x,u)+g(u)]h(x,u')\ in\ ]0, 1[\\
& u(0)=u(1)=0=u''(0)=u''(1), \tag{1}\end{align*} where \(\lambda\) is a positive parameter, \(f:[0, 1]\times\mathbb{R}\longrightarrow\mathbb{R}\) is an \(L^{1}-\)Caratheodory function, \(g:\mathbb{R}\longrightarrow\mathbb{R}\) is a Lipschitz continuous function and \(h:[0, 1]\times\mathbb{R}\longrightarrow [0,\infty[\) is a bounded and continuous function. Using variational methods and Ricceri's variational principle, they prove under suitable conditions on \(\lambda,f,g,h\) that problem (1) admits at least one nontrivial weak solution.Boundary control for optimal mixing via Navier-Stokes flowshttps://zbmath.org/1472.490052021-11-25T18:46:10.358925Z"Hu, Weiwei"https://zbmath.org/authors/?q=ai:hu.weiwei"Wu, Jiahong"https://zbmath.org/authors/?q=ai:wu.jiahongNecessary and sufficient conditions of optimality for a damped hyperbolic equation in one-space dimensionhttps://zbmath.org/1472.490062021-11-25T18:46:10.358925Z"Kucuk, Ismail"https://zbmath.org/authors/?q=ai:kucuk.ismail"Yildirim, Kenan"https://zbmath.org/authors/?q=ai:yildirim.kenanSummary: The present paper deals with the necessary optimality condition for a class of distributed parameter systems in which the system is modeled in one-space dimension by a hyperbolic partial differential equation subject to the damping and mixed constraints on state and controls. Pontryagin maximum principle is derived to be a necessary condition for the controls of such systems to be optimal. With the aid of some convexity assumptions on the constraint functions, it is obtained that the maximum principle is also a sufficient condition for the optimality.Optimal control for the infinity obstacle problemhttps://zbmath.org/1472.490072021-11-25T18:46:10.358925Z"Mawi, Henok"https://zbmath.org/authors/?q=ai:mawi.henok"Ndiaye, Cheikh Birahim"https://zbmath.org/authors/?q=ai:ndiaye.cheikh-birahimThe authors investigate an \(\infty\)-obstacle \(f\) optimal control problem \[ W_g^{1,2} (\Omega) = \{ u \in W^{1,2}(\Omega): \; u=g \; \text {on} \; \partial \Omega \} \] consists of minimizing the Dirichlet energy \( \int_\Omega |Du(x)|^2 dx\) over the set
\[ \mathbb{K}_{f,g}^2 = \{ u \in W^{1,2}(\Omega): \; u(x) \ge f(x) \; \text{in} \; \Omega \} \] where \(\Omega \in \mathbb{R}^n\) is a bounded and smooth domain, \(Du\) is the gradient of \(u\), and \(g \in \text{tr}(W^{1,2}(\Omega))\) with tr the trace operator. This optimal control problem models the equilibrium position of an elastic membrane whose boundary is held fixed at \(g\) and is forced to remain above the given obstacle \(f\). It was shown that the optimal control is also an optimal state. Moreover, it was proved that the minimal value of an optimal control problem for the finite obstacle converges to the minimal value of the optimal control problem for the infinite obstacle.Time-optimal control problem with state constraints in a time-periodic flow fieldhttps://zbmath.org/1472.490082021-11-25T18:46:10.358925Z"Chertovskih, Roman"https://zbmath.org/authors/?q=ai:chertovskih.roman-a"Khalil, Nathalie T."https://zbmath.org/authors/?q=ai:khalil.nathalie-t"Pereira, Fernando Lobo"https://zbmath.org/authors/?q=ai:lobo-pereira.fernandoSummary: The following time-optimal control problem is solved numerically: compute the fastest trajectory joining two given (initial and final) points of a dynamic control system in a time-periodic flow field subject to state constraints. The considered problem mimics the real-life task of path-planning of a ship in a flow with tidal variations. The considered problem is solved using the maximum principle in Gamkrelidze's form. Under reasonable assumptions on the flow field, it is proved, that the problem is regular and the measure Lagrange multiplier, associated with the state constraint, is continuous. These properties (regularity and continuity) play a critical role in computing the field of extremals by solving the two-point boundary value problem given by the maximum principle. Some examples of time-periodic fluid flows are considered and the corresponding optimal solutions are found.
For the entire collection see [Zbl 1460.90005].Singular optimal control problems for doubly nonlinear and quasi-variational evolution equationshttps://zbmath.org/1472.490092021-11-25T18:46:10.358925Z"Kenmochi, Nobuyuki"https://zbmath.org/authors/?q=ai:kenmochi.nobuyuki"Shirakawa, Ken"https://zbmath.org/authors/?q=ai:shirakawa.ken"Yamazaki, Noriaki"https://zbmath.org/authors/?q=ai:yamazaki.noriakiSummary: Doubly nonlinear and quasi-variational evolution equations governed by double time-dependent subdifferentials are treated in uniformly convex Banach spaces. We establish some abstract results on the existence-uniqueness of solutions together with related optimal control problems in cases when, in general, the state equations have multiple solutions. In this paper, we propose a general class of singular optimal control problems that are set up for non-well-posed state systems. Moreover, we establish an approximation procedure for such singular optimal control problems and discuss some applications.Existence results for strong mixed vector equilibrium problem for multivalued mappingshttps://zbmath.org/1472.490102021-11-25T18:46:10.358925Z"Kılıçman, Adem"https://zbmath.org/authors/?q=ai:kilicman.adem"Ahmad, Rais"https://zbmath.org/authors/?q=ai:ahmad.rais"Rahaman, Mijanur"https://zbmath.org/authors/?q=ai:rahaman.mijanurSummary: We consider a strong mixed vector equilibrium problem in topological vector spaces. Using generalized Fan-Browder fixed point theorem (Takahashi 1976) and generalized pseudomonotonicity for multivalued mappings, we provide some existence results for strong mixed vector equilibrium problem without using KKM-Fan theorem. The results in this paper generalize, improve, extend, and unify some existence results in the literature. Some special cases are discussed and an example is constructed.Error analysis through energy minimization and stability properties of exponential integratorshttps://zbmath.org/1472.490112021-11-25T18:46:10.358925Z"Kosmas, Odysseas"https://zbmath.org/authors/?q=ai:kosmas.odysseas"Vlachos, Dimitrios"https://zbmath.org/authors/?q=ai:vlachos.dimitriosSummary: In this article, the stability property and the error analysis of higher-order exponential variational integration are examined and discussed. Toward this purpose, at first we recall the derivation of these integrators and then address the eigenvalue problem of the amplification matrix for advantageous choices of the number of intermediate points employed. Obviously, the latter determines the order of the numerical accuracy of the method. Following a linear stability analysis process we show that the methods with at least one intermediate point are unconditionally stable. Finally, we explore the behavior of the energy errors of the presented schemes in prominent numerical examples and point out their excellent efficiency in long term integration.
For the entire collection see [Zbl 1470.49002].Infinite horizon discrete-time nonautonomous problems on axishttps://zbmath.org/1472.490122021-11-25T18:46:10.358925Z"Zaslavski, Alexander J."https://zbmath.org/authors/?q=ai:zaslavski.alexander-jSummary: In this paper we discuss our recent results on turnpike conditions and establish the existence of solutions over infinite horizon for discrete-time nonautonomous problems on axis in metric spaces.Solvability and optimal controls of semilinear Riemann-Liouville fractional differential equationshttps://zbmath.org/1472.490132021-11-25T18:46:10.358925Z"Pan, Xue"https://zbmath.org/authors/?q=ai:pan.xue"Li, Xiuwen"https://zbmath.org/authors/?q=ai:li.xiuwen"Zhao, Jing"https://zbmath.org/authors/?q=ai:zhao.jing.1|zhao.jing.2|zhao.jing|zhao.jing.3Summary: We consider the control systems governed by semilinear differential equations with Riemann-Liouville fractional derivatives in Banach spaces. Firstly, by applying fixed point strategy, some suitable conditions are established to guarantee the existence and uniqueness of mild solutions for a broad class of fractional infinite dimensional control systems. Then, by using generally mild conditions of cost functional, we extend the existence result of optimal controls to the Riemann-Liouville fractional control systems. Finally, a concrete application is given to illustrate the effectiveness of our main results.The property of the set of equilibria of the equilibrium problem with lower and upper bounds on Hadamard manifoldshttps://zbmath.org/1472.490142021-11-25T18:46:10.358925Z"Zhang, Qing-Bang"https://zbmath.org/authors/?q=ai:zhang.qingbang"Tang, Gusheng"https://zbmath.org/authors/?q=ai:tang.gushengSummary: The existence of equilibrium points, and the essential stability of the set of equilibrium points of the equilibrium problem with lower and upper bounds are studied on Hadamard manifolds.Existence theorems and regularization methods for non-coercive vector variational and vector quasi-variational inequalitieshttps://zbmath.org/1472.490152021-11-25T18:46:10.358925Z"Khan, Akhtar A."https://zbmath.org/authors/?q=ai:khan.akhtar-ali"Hebestreit, Niklas"https://zbmath.org/authors/?q=ai:hebestreit.niklas"Köbis, Elisabeth"https://zbmath.org/authors/?q=ai:kobis.elisabeth"Tammer, Christiane"https://zbmath.org/authors/?q=ai:tammer.christianeSummary: In this paper, we present a novel existence result for vector variational inequalities with respect to a fixed ordering cone. Since these problems are ill-posed in general, we further propose a regularization technique for non-coercive problems which allows to derive existence statements even when the common coercivity conditions in the literature do not hold. For this purpose, we replace our original problem by a family of well-behaving problems and study their relationships. In order to justify our abstract framework we apply our results to generalized vector variational inequalities and multi-objective optimization problems. We further consider vector quasi-variational inequalities and prove existence of solutions using a regularization approach and a fixed-point theorem for single-valued mappings.Uzawa-type iterative solution methods for constrained saddle point problemshttps://zbmath.org/1472.490162021-11-25T18:46:10.358925Z"Lapin, A."https://zbmath.org/authors/?q=ai:lapin.aleksandr-vasilevichThe paper deals with the iterative solution of saddle point problems in finite dimensional real valued spaces. It summarizes the general theory developed by the author with co-authors in the last years. For finite-dimensional saddle point problem with a nonlinear monotone operator and constraints, iterative methods are knows, which can be viewed as preconditioned Uzawa methods or as Uzawa-block relaxation methods. In the first section the author gives some general results on the solvability of problem the considered problem and the convergence of iterative methods. These results are based on three articles from the author and coworkers from the last years. In the following sections some application on the finite element/finete differences discretization of elliptic and parabolic PDEs optimal control problems and the resulting (optimality) conditions, i.e. saddle point problems are considered. When applying iterative method to such problems, one problem is to construct suitable preconditioners that ensure the convergence and effective implementation of iterative methods and to obtain the admissible intervals of iterative parameters which do not depend on mesh parameters. The choice of preconditioners is handled case by case for the considered example problems.Split generalized vector variational inequalities for set-valued mappings and applications to social utility optimizations with uncertaintyhttps://zbmath.org/1472.490172021-11-25T18:46:10.358925Z"Li, Jinlu"https://zbmath.org/authors/?q=ai:li.jinlu"Stone, Glenn"https://zbmath.org/authors/?q=ai:stone.glenn-davisSummary: In this paper, we use the power, upward power and the downward power preorders on the power sets of topological vector spaces to define the split generalized vector variational inequality problems for set-valued mappings on topological vector spaces. By using the Fan-KKM theorem, we prove an existence theorem for solutions to some split generalized vector variational inequality problems for set-valued mappings on topological vector spaces. Consequently, we prove the solvability of some generalized vector variational inequality problems for set-valued mappings. As applications, we study the existence of efficient pair of initial social activity profiles for some split generalized social utility optimization problems.Solvability of ordered equilibrium problems on partially ordered spaceshttps://zbmath.org/1472.490182021-11-25T18:46:10.358925Z"Li, Jinlu"https://zbmath.org/authors/?q=ai:li.jinlu"Wen, Ching-Feng"https://zbmath.org/authors/?q=ai:wen.chingfengSummary: The concept of ordered equilibrium problems on partially ordered sets extends the concept of vector equilibrium problems on topological vector spaces, while the latter is a natural extension of equilibrium problems on topological vector spaces. In this paper, for generalizing the semicontinuity and concavity of real valued functions, we introduce the concepts of ordered semicontinuity and ordered concavity for mappings that take values in partially ordered spaces. We prove some existence theorems for solutions of some ordered equilibrium problems by using the Fan-KKM theorem. Then we apply this theorem to show the solvability of some vector equilibrium problems and some ordinary equilibrium problems.Infinitely nonlinear split variational inequalities in Banach spaceshttps://zbmath.org/1472.490192021-11-25T18:46:10.358925Z"Li, Jinlu"https://zbmath.org/authors/?q=ai:li.jinlu"Xie, Linsen"https://zbmath.org/authors/?q=ai:xie.linsen"Zhao, Xiaopeng"https://zbmath.org/authors/?q=ai:zhao.xiaopengSummary: In this paper, we introduce the concept of infinitely nonlinear split variational inequality problems on Banach spaces which includes infinitely linear split variational inequality problems, finitely nonlinear split variational inequality problems and split variational inequality problems as special cases. We also introduce the concepts of semicontinuity and concavity of mappings with respect to other operators and the concepts of convexly order reserved mappings related with other two or three mappings. By using the Fan-KKM theorem, we prove the existence of solutions to some infinitely nonlinear split variational inequality problems on Banach spaces.Existence of solutions and algorithm for generalized vector quasi-complementarity problems with application to traffic network problemshttps://zbmath.org/1472.490202021-11-25T18:46:10.358925Z"Nguyen Van Hung"https://zbmath.org/authors/?q=ai:nguyen-van-hung."Vo Minh Tam"https://zbmath.org/authors/?q=ai:vo-minh-tam."Köbis, Elisabeth"https://zbmath.org/authors/?q=ai:kobis.elisabeth"Yao, Jen-Chih"https://zbmath.org/authors/?q=ai:yao.jen-chihSummary: In this paper, we establish a new class of generalized vector quasi complementarity problems with fuzzy mappings in Hausdorff topological vector spaces. Afterward, we prove that generalized vector quasi-complementarity problems with fuzzy mappings are equivalent to generalized vector quasi-variational inequality problems with fuzzy mappings by using suitable conditions. The existence of solutions for our problems by using the Kakutani-Fan-Glicksberg fixed point theorem is obtained. Further, based on an auxiliary problem, we propose a projection iterative algorithm for generalized vector variational inequality problems with fuzzy mappings. Under suitable conditions, we prove the convergence of this iterative algorithm and solve the generalized vector complementarity problem with fuzzy mappings. As a real-world application, we consider the special case of traffic network problems. A part of the main results obtained in this paper represents an affirmative answer to an open problem posed in [\textit{A. Kılıçman} et al., Fuzzy Sets Syst. 280, 133--141 (2015; Zbl 1378.49009)].Correction to: ``On nonlocal variational and quasi-variational inequalities with fractional gradient''https://zbmath.org/1472.490212021-11-25T18:46:10.358925Z"Rodrigues, José Francisco"https://zbmath.org/authors/?q=ai:rodrigues.jose-francisco"Santos, Lisa"https://zbmath.org/authors/?q=ai:santos.lisaCorrection to the authors' paper [ibid. 80, No. 3, 835--852 (2019; Zbl 1429.49011)].Constrained weak Nash-type equilibrium problemshttps://zbmath.org/1472.490222021-11-25T18:46:10.358925Z"Shuai, W. C."https://zbmath.org/authors/?q=ai:shuai.w-c"Xiang, K. L."https://zbmath.org/authors/?q=ai:xiang.kaili"Zhang, W. Y."https://zbmath.org/authors/?q=ai:zhang.weiyu|zhang.weiyi.1|zhang.wenying|zhang.wenyang|zhang.weiyong|zhang.wenyi|zhang.wenyue|zhang.weiyi.2|zhang.wenyan|zhang.weiye|zhang.wanying|zhang.weiya|zhang.wenyong|zhang.wenyin|zhang.wenyuan|zhang.wenyu|zhang.weiyuan|zhang.wen-yun|zhang.wenyao|zhang.wuyangSummary: A constrained weak Nash-type equilibrium problem with multivalued payoff functions is introduced. By virtue of a nonlinear scalarization function, some existence results are established. The results extend the corresponding one of \textit{J.-Y. Fu} [J. Math. Anal. Appl. 285, No. 2, 708--713 (2003; Zbl 1031.49013)]. In particular, if the payoff functions are singlevalued, our existence theorem extends the main results of [loc. cit.] by relaxing the assumption of convexity.Semicontinuity of the solution mapping to a parametric generalized weak Ky Fan inequalityhttps://zbmath.org/1472.490232021-11-25T18:46:10.358925Z"Xu, Y. D."https://zbmath.org/authors/?q=ai:xu.yidan|xu.yongdeng|xu.yangdong|xu.yongdong|xu.you-dong|xu.yuedong|xu.yuandong|xu.yundou|xu.yadong|xu.yida|xu.yidongSummary: Under new assumptions, which do not contain any information about the solution set, the upper and lower semicontinuity of the solution mapping to a class of parametric generalized weak Ky Fan inequality are established by using a nonlinear scalarization technique. These results extend and improve the recent ones in the literature. Some examples are given to illustrate our results.Nonemptiness and compactness of solution sets to weakly homogeneous generalized variational inequalitieshttps://zbmath.org/1472.490242021-11-25T18:46:10.358925Z"Zheng, Meng-Meng"https://zbmath.org/authors/?q=ai:zheng.mengmeng"Huang, Zheng-Hai"https://zbmath.org/authors/?q=ai:huang.zheng-hai"Bai, Xue-Li"https://zbmath.org/authors/?q=ai:bai.xueliSummary: In this paper, we deal with the weakly homogeneous generalized variational inequality, which provides a unified setting for several special variational inequalities and complementarity problems studied in recent years. By exploiting weakly homogeneous structures of involved map pairs and using degree theory, we establish a result which demonstrates the connection between weakly homogeneous generalized variational inequalities and weakly homogeneous generalized complementarity problems. Subsequently, we obtain a result on the nonemptiness and compactness of solution sets to weakly homogeneous generalized variational inequalities by utilizing Harker-Pang-type condition, which can lead to a Hartman-Stampacchia-type existence theorem. Last, we give several copositivity results for weakly homogeneous generalized variational inequalities, which can reduce to some existing ones.The Tikhonov regularization for vector equilibrium problemshttps://zbmath.org/1472.490252021-11-25T18:46:10.358925Z"Anh, Lam Quoc"https://zbmath.org/authors/?q=ai:anh.lam-quoc"Duy, Tran Quoc"https://zbmath.org/authors/?q=ai:duy.tran-quoc"Muu, Le Dung"https://zbmath.org/authors/?q=ai:muu.le-dung"Tri, Truong Van"https://zbmath.org/authors/?q=ai:tri.truong-vanAuthors' abstract: We consider vector equilibrium problems in real Banach spaces and study their regularized problems. Based on cone continuity and generalized convexity properties of vector-valued mappings, we propose general conditions that guarantee existence of solutions to such problems in cases of monotonicity and nonmonotonicity. First, our study indicates that every Tikhonov trajectory converges to a solution to the original problem. Then, we establish the equivalence between the problem solvability and the boundedness of any Tikhonov trajectory. Finally, the convergence of the Tikhonov trajectory to the least-norm solution of the original problem is discussed.Double phase image restorationhttps://zbmath.org/1472.490262021-11-25T18:46:10.358925Z"Harjulehto, Petteri"https://zbmath.org/authors/?q=ai:harjulehto.petteri"Hästö, Peter"https://zbmath.org/authors/?q=ai:hasto.peter-aThe double phase functional was introduced by Zhikov in 1980 and intensively studied by many authors after 2015 in different aspects as double phase problems, regularity of solutions, Calderon-Zigmund estimets, etc. In the present article, the authors study the potential of double phase functional in the image processing domain. The functional is considered in the space BV of functions of bounded variation. A detailed Introduction is given, followed by two sections of preliminaries on the functional and BV spaces. The main result is Theorem 4.1 formulated and proved in Section 4. It is for \(\Gamma\)-convergence of considered functional in space \(L^1\).A Lagrange multiplier method for semilinear elliptic state constrained optimal control problemshttps://zbmath.org/1472.490272021-11-25T18:46:10.358925Z"Karl, Veronika"https://zbmath.org/authors/?q=ai:karl.veronika"Neitzel, Ira"https://zbmath.org/authors/?q=ai:neitzel.ira"Wachsmuth, Daniel"https://zbmath.org/authors/?q=ai:wachsmuth.danielAuthors' abstract: In this paper we apply an augmented Lagrange method to a class of semilinear elliptic optimal control problems with pointwise state constraints. We show strong convergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak-star convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.A predictor-corrector method for solving equilibrium problemshttps://zbmath.org/1472.490282021-11-25T18:46:10.358925Z"Bao, Zong-Ke"https://zbmath.org/authors/?q=ai:bao.zong-ke"Huang, Ming"https://zbmath.org/authors/?q=ai:huang.ming"Xia, Xi-Qiang"https://zbmath.org/authors/?q=ai:xia.xiqiangSummary: We suggest and analyze a predictor-corrector method for solving nonsmooth convex equilibrium problems based on the auxiliary problem principle. In the main algorithm each stage of computation requires two proximal steps. One step serves to predict the next point; the other helps to correct the new prediction. At the same time, we present convergence analysis under perfect foresight and imperfect one. In particular, we introduce a stopping criterion which gives rise to \(\Delta\)-stationary points. Moreover, we apply this algorithm for solving the particular case: variational inequalities.Qualification conditions-free characterizations of the \(\varepsilon \)-subdifferential of convex integral functionshttps://zbmath.org/1472.490292021-11-25T18:46:10.358925Z"Correa, Rafael"https://zbmath.org/authors/?q=ai:correa.rafael"Hantoute, Abderrahim"https://zbmath.org/authors/?q=ai:hantoute.abderrahim"Pérez-Aros, Pedro"https://zbmath.org/authors/?q=ai:perez-aros.pedroSummary: We provide formulae for the \(\varepsilon \)-subdifferential of the integral function \(I_f(x):=\int_T f(t,x)\, d\mu (t)\), where the integrand \(f:T\times X \rightarrow \overline{\mathbb{R}}\) is measurable in \((t, x)\) and convex in \(x\). The state variable lies in a locally convex space, possibly non-separable, while \(T\) is given a structure of a nonnegative complete \(\sigma \)-finite measure space \((T,\mathcal{A},\mu )\). The resulting characterizations are given in terms of the \(\varepsilon \)-subdifferential of the data functions involved in the integrand, \(f\), without requiring any qualification conditions. We also derive new formulas when some usual continuity-type conditions are in force. These results are new even for the finite sum of convex functions and for the finite-dimensional setting.Erratum to: ``Optimality conditions for nonsmooth interval-valued and multiobjective semi-infinite programming''https://zbmath.org/1472.490302021-11-25T18:46:10.358925Z"Jennane, Mohsine"https://zbmath.org/authors/?q=ai:jennane.mohsine"Kalmoun, El Mostafa"https://zbmath.org/authors/?q=ai:kalmoun.el-mostafa"Lafhim, Lahoussine"https://zbmath.org/authors/?q=ai:lafhim.lahoussineSummary: This note corrects an error in our paper [ibid. 55, No. 1, 1--11 (2021; Zbl 1468.49016)] as we should drop the expression ``\textit{with at least one strict inequality}'' in the definition of interval order in Section 2. Instead of proposing this short amendment, \textit{N. A. Gadhi} and \textit{A. Ichatouhane} [ibid. 55, No. 1, 13--22 (2021; Zbl 1468.49014)] gave a proposition that requires an additional condition on the constraint functions. However, we claim that all the results of our paper are correct once the modification above is done.On nonsmooth semi-infinite minimax programming problem with \((\Phi, \rho)\)-invexityhttps://zbmath.org/1472.490312021-11-25T18:46:10.358925Z"Liu, X. L."https://zbmath.org/authors/?q=ai:liu.xiaoling"Lai, G. M."https://zbmath.org/authors/?q=ai:lai.guoming"Xu, C. Q."https://zbmath.org/authors/?q=ai:xu.chuanqing"Yuan, D. H."https://zbmath.org/authors/?q=ai:yuan.dehuiSummary: We are interested in a nonsmooth minimax programming Problem (SIP). Firstly, we establish the necessary optimality conditions theorems for Problem (SIP) when using the well-known Caratheodory's theorem. Under the Lipschitz \((\Phi, \rho)\)-invexity assumptions, we derive the sufficiency of the necessary optimality conditions for the same problem. We also formulate dual and establish weak, strong, and strict converse duality theorems for Problem (SIP) and its dual. These results extend several known results to a wider class of problems.On continuous codifferentiability of quasidifferentiable functionshttps://zbmath.org/1472.490322021-11-25T18:46:10.358925Z"Prudnikov, Igor M."https://zbmath.org/authors/?q=ai:prudnikov.igor-mSummary: The author studied codifferentiable functions introduced by Professor V. F. Demyanov and a method for calculating their subdifferentials and codifferentials. The proven theorems give the rules for calculating the subdifferentials. It was shown that the subdifferential of the first order, introduced by the author, coincides with the Demyanov's difference applied to the subdifferential and superdifferential of a Lipschitz quasidifferentiable function. The author proved that any quasidifferentiable function is continuously codifferentiable under the condition of upper semicontinuity for the subdifferential and superdifferential mappings. The author suggested constructing a continuous extension for the subdifferential of the first order and introduced \(\alpha \) -optimal points.Approximate solutions in nonsmooth robust optimization with locally Lipschitz constraintshttps://zbmath.org/1472.490332021-11-25T18:46:10.358925Z"Shitkovskaya, Tatiana"https://zbmath.org/authors/?q=ai:shitkovskaya.tatiana"Hong, Zhe"https://zbmath.org/authors/?q=ai:hong.zhe"Jiao, Liguo"https://zbmath.org/authors/?q=ai:jiao.liguo"Kim, Do Sang"https://zbmath.org/authors/?q=ai:kim.do-sangSummary: In this paper, we consider a robust optimization problem with locally Lipschitz constraints, and study some characterizations of a so-called quasi \(\epsilon\)-solution in this setting. In other words, a necessary optimality condition for a quasi \(\epsilon\)-solution to the mentioned problem is established under some suitable robust constraint qualification; sufficient optimality conditions are also provided with the help of generalized convexity. The optimality conditions are presented in terms of multipliers and limiting subdifferentials of the related functions. Moreover, a simple example is provided to illustrate necessary optimality conditions. Finally, approximate duality relationships are discussed after a Wolfe type dual model (in approximate form) is formulated.Generalized error bound for conic inequality in \(\mathfrak{C}^2\) type Banach spaceshttps://zbmath.org/1472.490342021-11-25T18:46:10.358925Z"Zhu, Jiangxing"https://zbmath.org/authors/?q=ai:zhu.jiangxing"Hu, Chunhai"https://zbmath.org/authors/?q=ai:hu.chunhai"Ouyang, Wei"https://zbmath.org/authors/?q=ai:ouyang.weiSummary: In this paper, we consider generalized error bound issues for conic inequalities in \(\mathfrak{C}^2\) type Banach spaces. Using the techniques of variational analysis and in terms of proximal subdifferentials of vector-valued functions, we provide sufficient conditions for the existence of a generalized error bound for a conic inequality. In particular, our results improve and extend some existing results.Ekeland type variational principle for set-valued maps in quasi-metric spaces with applicationshttps://zbmath.org/1472.490352021-11-25T18:46:10.358925Z"Ansari, Qamrul Hasan"https://zbmath.org/authors/?q=ai:ansari.qamrul-hasan"Sharma, Pradeep Kumar"https://zbmath.org/authors/?q=ai:sharma.pradeep-kumarSummary: In this paper, we derive a fixed point theorem, minimal element theorems and Ekeland type variational principle for set-valued maps with generalized variable set relations in quasi-metric spaces. These generalized variable set relations are the generalizations of set relations with constant ordering cone, and form the modern approach to compare sets in set-valued optimization with respect to variable domination structures under some appropriate assumptions. At the end, we give application of these variational principles to the capability theory of well-beings via variational rationality.Generalized sequential differential calculus for expected-integral functionalshttps://zbmath.org/1472.490362021-11-25T18:46:10.358925Z"Mordukhovich, Boris S."https://zbmath.org/authors/?q=ai:mordukhovich.boris-s"Pérez-Aros, Pedro"https://zbmath.org/authors/?q=ai:perez-aros.pedroIntegral functionals depending on a parameter of the type
\[
E(x, y) :=\int_T \varphi_t (x,y(t)) \mu (dt)
\]
are considered, where \(x\in \mathbb{R}^n\) and \(y\in L^1(T;\mathbb{R}^m)\), with \(T\) a measure space. The main assumptions on \(\varphi\) are a bound from below and convexity with respect to \(y\). The point of the paper is representing elements of the subdifferential of \(E\) at \((\bar x,\bar y)\) as a limit of integrals involving the subdifferential of \(\varphi\) at suitable nearby points. In particular, the point \(\bar x\) needs to be approximated by an \(L^p\) function in a suitable way.Regularity of implicit solution mapping to parametric generalized equationhttps://zbmath.org/1472.490372021-11-25T18:46:10.358925Z"Ouyang, Wei"https://zbmath.org/authors/?q=ai:ouyang.wei"Zhang, Binbin"https://zbmath.org/authors/?q=ai:zhang.binbinSummary: This paper concerns the study of both local and global metric regularity/Lipschitz-like properties concerning the behavior of the implicit solution mapping associated to parametric generalized equation in metric space. We extend some implicit multifunction results to the addition of two multifunctions both depending on parameters. Through the approach of inverse mapping iteration, several results are established regarding the relations between the (partial) metric regularity/Lipschitz-like moduli of multifunctions used as the defining form of the generalized equation and the corresponding implicit solution mapping, the proof of which is completely self-contained. Finally, a local Lyusternik-Graves Theorem is obtained as an application.Applying twice a minimax theoremhttps://zbmath.org/1472.490382021-11-25T18:46:10.358925Z"Ricceri, Biagio"https://zbmath.org/authors/?q=ai:ricceri.biagioSummary: Here is one of the results obtained in this paper: Let \(X\), \(Y\) be two convex sets each in a real vector space, let \(J:X\times Y\to\mathbb{R}\) be convex and without global minima in \(X\) and concave in \(Y\), and let \(\Phi:X\to\mathbb{R}\) be strictly convex. Also, assume that, for some topology on \(X\), \(\Phi\) is lower semicontinuous and, for each \(y\in Y\) and \(\lambda>0\), \(J(\cdot,y)\) is lower semicontinuous and \(J(\cdot,y)+\lambda\Phi(\cdot)\) is inf-compact.
Then, for each \(r\in]\inf_X\Phi,\sup_X\Phi[\) and for each closed set \(S\subseteq X\) satisfying
\[
\Phi^{-1}(r)\subseteq S\subseteq\Phi^{-1}(]-\infty,r]),
\]
one has
\[
\sup\limits_Y\inf\limits_S J=\inf\limits_S\sup\limits_Y J.
\]The optimal control problem with state constraints for fully coupled forward-backward stochastic systems with jumpshttps://zbmath.org/1472.490392021-11-25T18:46:10.358925Z"Wei, Qingmeng"https://zbmath.org/authors/?q=ai:wei.qingmengSummary: We focus on the fully coupled forward-backward stochastic differential equations with jumps and investigate the associated stochastic optimal control problem (with the nonconvex control and the convex state constraint) along with stochastic maximum principle. To derive the necessary condition (i.e., stochastic maximum principle) for the optimal control, first we transform the fully coupled forward-backward stochastic control system into a fully coupled backward one; then, by using the terminal perturbation method, we obtain the stochastic maximum principle. Finally, we study a linear quadratic model.An optimal control problem of forward-backward stochastic Volterra integral equations with state constraintshttps://zbmath.org/1472.490402021-11-25T18:46:10.358925Z"Wei, Qingmeng"https://zbmath.org/authors/?q=ai:wei.qingmeng"Xiao, Xinling"https://zbmath.org/authors/?q=ai:xiao.xinlingSummary: This paper is devoted to the stochastic optimal control problems for systems governed by forward-backward stochastic Volterra integral equations (FBSVIEs, for short) with state constraints. Using Ekeland's variational principle, we obtain one kind of variational inequalities. Then, by dual method, we derive a stochastic maximum principle which gives the necessary conditions for the optimal controls.Optimality conditions for the continuous model of the final open pit problemhttps://zbmath.org/1472.490412021-11-25T18:46:10.358925Z"Amaya, Jorge"https://zbmath.org/authors/?q=ai:amaya.jorge"Hermosilla, Cristopher"https://zbmath.org/authors/?q=ai:hermosilla.cristopher"Molina, Emilio"https://zbmath.org/authors/?q=ai:molina.emilioThe authors discuss optimality conditions for the ``final open pit problem''. Here they model a pit as a depth function \(p\) over some domain \(\Omega \subset \mathbb{R}^d\), to be determined, maximizing the gain obtained from excavating the pit. For with additional constraints on the function values of \(p\), its slope, and a bound on the associated costs of the excavation.
For this problem, the authors derive optimality conditions for the case \(d = 1\) and \(d =2\). In the case \(d=2\), the authors only provide conditions on subdomains \(\Omega_0 \subset \Omega\) on which the slope constraints remain inactive.A projected primal-dual gradient optimal control method for deep reinforcement learninghttps://zbmath.org/1472.490422021-11-25T18:46:10.358925Z"Gottschalk, Simon"https://zbmath.org/authors/?q=ai:gottschalk.simon"Burger, Michael"https://zbmath.org/authors/?q=ai:burger.michael"Gerdts, Matthias"https://zbmath.org/authors/?q=ai:gerdts.matthiasSummary: In this contribution, we start with a policy-based Reinforcement Learning ansatz using neural networks. The underlying Markov Decision Process consists of a transition probability representing the dynamical system and a policy realized by a neural network mapping the current state to parameters of a distribution. Therefrom, the next control can be sampled. In this setting, the neural network is replaced by an ODE, which is based on a recently discussed interpretation of neural networks. The resulting infinite optimization problem is transformed into an optimization problem similar to the well-known optimal control problems. Afterwards, the necessary optimality conditions are established and from this a new numerical algorithm is derived. The operating principle is shown with two examples. It is applied to a simple example, where a moving point is steered through an obstacle course to a desired end position in a 2D plane. The second example shows the applicability to more complex problems. There, the aim is to control the finger tip of a human arm model with five degrees of freedom and 29 Hill's muscle models to a desired end position.Optimization-constrained differential equations with active set changeshttps://zbmath.org/1472.490432021-11-25T18:46:10.358925Z"Stechlinski, Peter"https://zbmath.org/authors/?q=ai:stechlinski.peter-gThis paper deals with parameterized DAEs featuring algebraic inequalities (not only exact constraints), for which solutions should realize an optimization objective. Upon topological and metric conditions, the authors build the theory of such problems: solution existence, uniqueness and dependency to the parameters. The theory gives regularities conditions for such a system to have a generalized index one (Theorem 3.1) and adapt them to the parameterized optimization problem (Theorem 4.1). The content is quite notational though comprehensive, and well organized. It is claimed to be computationally relevant, in that the use of lexigographic directional derivative helps to solve the optimization condition, locally. Really useful for experts of the field, this appealing results need additional examples from real control systems (as mentioned in the introduction) to help right understanding.Optimal control of a non-smooth quasilinear elliptic equationhttps://zbmath.org/1472.490442021-11-25T18:46:10.358925Z"Clason, Christian"https://zbmath.org/authors/?q=ai:clason.christian"Nhu, Vu Huu"https://zbmath.org/authors/?q=ai:nhu.vu-huu"Rösch, Arnd"https://zbmath.org/authors/?q=ai:rosch.arndSummary: This work is concerned with an optimal control problem governed by a non-smooth quasilinear elliptic equation with a nonlinear coefficient in the principal part that is locally Lipschitz continuous and directionally but not Gâteaux differentiable. This leads to a control-to-state operator that is directionally but not Gâteaux differentiable as well. Based on a suitable regularization scheme, we derive C- and strong stationarity conditions. Under the additional assumption that the nonlinearity is a \(PC^1\) function with countably many points of nondifferentiability, we show that both conditions are equivalent. Furthermore, under this assumption we derive a relaxed optimality system that is amenable to numerical solution using a semi-smooth Newton method. This is illustrated by numerical examples.On the supremal version of the Alt-Caffarelli minimization problemhttps://zbmath.org/1472.490452021-11-25T18:46:10.358925Z"Crasta, Graziano"https://zbmath.org/authors/?q=ai:crasta.graziano"Fragalà, Ilaria"https://zbmath.org/authors/?q=ai:fragala.ilariaThe authors consider the free boundary problem (P)\(_\Lambda\): \[m_\Lambda:=\min\{J_\Lambda(u):=\|\nabla u\|_\infty+\Lambda|\{u>0\}|:u\in \text{Lip}_1(\Omega)\},\] where \(\Lambda>0,\ |\{u>0\}|\) is the Lebesgue measure of the set \(\{x\in \Omega:u(x)>0\}\) and \[\text{Lip}_1(\Omega):= \{u\in W^{1,\infty}(\Omega)\,:u\ge 0\ \text{in}\ \Omega, \ u=1\ \text{on}\ \partial\Omega\}\] The main results include the existence and uniqueness of the non-constant solution on convex domains, the identification and the geometrical characterization of the variational infinity Bernoulli constant \[\Lambda_{\Omega,\infty}:=\inf\{\Lambda>0:\,(P)_\Lambda\ \text{admits a non-constant solution}\}.\]Meshless methods for solving Dirichlet boundary optimal control problems governed by elliptic PDEshttps://zbmath.org/1472.490462021-11-25T18:46:10.358925Z"Guan, Hongbo"https://zbmath.org/authors/?q=ai:guan.hongbo"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong|wang.yong.7|wang.yong.9|wang.yong.5|wang.yong.6|wang.yong.3|wang.yong.8|wang.yong.10|wang.yong.1|wang.yong.2"Zhu, Huiqing"https://zbmath.org/authors/?q=ai:zhu.huiqingSummary: In this paper, two meshless schemes are proposed for solving Dirichlet boundary optimal control problems governed by elliptic equations. The first scheme uses radial basis function collocation method (RBF-CM) for both state equation and adjoint state equation, while the second scheme employs the method of fundamental solution (MFS) for the state equation when it has a zero source term, and RBF-CM for the adjoint state equation. Numerical examples are provided to validate the efficiency of the proposed schemes.Comparison of direct and indirect approaches for numerical solution of the optimal control problem by evolutionary methodshttps://zbmath.org/1472.490472021-11-25T18:46:10.358925Z"Diveev, Askhat"https://zbmath.org/authors/?q=ai:diveev.askhat"Shmalko, Elizaveta"https://zbmath.org/authors/?q=ai:shmalko.elizavetaSummary: The optimal control problem with phase constraints is considered. A new indirect approach of synthesized optimal control is proposed as an alternative to direct methods. A comparative study of direct and indirect approaches is carried out on the problem of optimal control for a small group of mobile robots in the complex environment with phase constraints by evolutionary algorithms. With a direct approach to the numerical solution of the optimal control problem, the control function is searched in the form of piece-wise functional approximation. The indirect approach of synthesized optimal control comes from the engineering practice. Instead of reducing the optimal control problem to the problem of finite-dimensional optimization, we firstly make the object stable relative to some point in the state space by solving an additional task of synthesis of stabilizing control and then we find the coordinates of stabilization points as the desired parameters of optimal control.
For the entire collection see [Zbl 1460.90005].Optimal control of ODEs with state supremahttps://zbmath.org/1472.490482021-11-25T18:46:10.358925Z"Geiger, Tobias"https://zbmath.org/authors/?q=ai:geiger.tobias"Wachsmuth, Daniel"https://zbmath.org/authors/?q=ai:wachsmuth.daniel"Wachsmuth, Gerd"https://zbmath.org/authors/?q=ai:wachsmuth.gerdSummary: We consider the optimal control of a differential equation that involves the suprema of the state over some part of the history. In many applications, this non-smooth functional dependence is crucial for the successful modeling of real-world phenomena. We prove the existence of solutions and show that related problems may not possess optimal controls. Due to the non-smoothness in the state equation, we cannot obtain optimality conditions via standard theory. Therefore, we regularize the problem via a LogIntExp functional which generalizes the well-known LogSumExp. By passing to the limit with the regularization, we obtain an optimality system for the original problem. The theory is illustrated by some numerical experiments.Periodic solutions to the optimal control problem of rotation of a rigid body using internal masshttps://zbmath.org/1472.490492021-11-25T18:46:10.358925Z"Shmatkov, A. M."https://zbmath.org/authors/?q=ai:shmatkov.a-mSummary: A two-dimensional problem of time-optimal rotation of a mechanical system consisting of a rigid body and a mass point is considered. The mass point interacts with the body by internal forces only. The periodic optimal trajectories of the mass point passing through the rigid body center of inertia are found.Nondifferentiable minimax programming problem with second-order \((p,r)\)-invex functionshttps://zbmath.org/1472.490502021-11-25T18:46:10.358925Z"Antczak, Tadeusz"https://zbmath.org/authors/?q=ai:antczak.tadeusz"Slimani, Hachem"https://zbmath.org/authors/?q=ai:slimani.hachemSummary: In this paper, new generalized invexity concepts for twice differentiable functions are introduced. The definition of so-called second-order \((p,r)\)-invexity and its various generalizations are given and they are used for the considered nondifferentiable minimax programming problem. For such a nonsmooth optimization problem, second-order dual problems in the format of Wolfe and in the format of Mond-Weir are formulated. Thus, several second-order duality results are established under assumptions that the functions constituting the considered nondifferentiable minimax programming problem are second-order \((p,r)\)-invex and/or generalized second-order \((p,r)\)-invex functions. With the reference to the said functions, we extend some results of second-order dualty for a new class of nondifferentiable minimax programming problems with nonconvex twice differentiable functions.On sensitivity of vector equilibria by means of the diagonal subdifferential operatorhttps://zbmath.org/1472.490512021-11-25T18:46:10.358925Z"Al-Homidan, Suliman"https://zbmath.org/authors/?q=ai:al-homidan.suliman-s"Ansari, Qamrul Hasan"https://zbmath.org/authors/?q=ai:ansari.qamrul-hasan"Kassay, Gabor"https://zbmath.org/authors/?q=ai:kassay.gaborSummary: Based on the concept of subdifferential of a convex vector function, we define the so-called diagonal subdifferential operator for vector-valued bifunctions depending on a parameter and show its sensitivity with respect to the parameter. As a byproduct, we obtain Lipschitz continuity results of the solution map for parametric strong vector equilibrium problems.Equivalence results of well-posedness for split variational-hemivariational inequalitieshttps://zbmath.org/1472.490522021-11-25T18:46:10.358925Z"Hu, Rong"https://zbmath.org/authors/?q=ai:hu.rong"Xiao, Yi-Bin"https://zbmath.org/authors/?q=ai:xiao.yibin"Huang, Nan-Jing"https://zbmath.org/authors/?q=ai:huang.nan-jing"Wang, Xing"https://zbmath.org/authors/?q=ai:wang.xingSummary: In this paper, with the concept of well-posedness for the split hemivariational inequalities in the literature, we study a split variational-hemivariational inequality in reflexive Banach space. After introducing several concepts of well-posedness for the split variational-hemivariational inequality and the corresponding split inclusion problem, we establish some equivalence results of well-posedness between the two problems. Then, under different conditions, we prove the strong and weak well-posedness for the split variational-hemivariational inequality are equivalent to the existence and uniqueness of its solution, respectively.Painlevé-Kuratowski stability of the approximate solution sets for set-valued vector equilibrium problemshttps://zbmath.org/1472.490532021-11-25T18:46:10.358925Z"Peng, Zai Yun"https://zbmath.org/authors/?q=ai:peng.zaiyun"Zhao, Yong"https://zbmath.org/authors/?q=ai:zhao.yong"Yang, Xin Min"https://zbmath.org/authors/?q=ai:yang.xinminSummary: The aim of this paper is to study the stability aspects of approximate solution sets of perturbed vector equilibrium problems in Hausdorff topological vector spaces. Using a scalarization method, we establish a sufficient condition for Painlevé-Kuratowski convergence of \(\epsilon\)-approximate solutions set to set-valued vector equilibrium problems, where the sequence of mappings converge in the sense of \(\Gamma_C\). These results extend and improve some results in the literature. Some examples are given to illustrate the results.Viscosity solutions for controlled McKean-Vlasov jump-diffusionshttps://zbmath.org/1472.490542021-11-25T18:46:10.358925Z"Burzoni, Matteo"https://zbmath.org/authors/?q=ai:burzoni.matteo"Ignazio, Vincenzo"https://zbmath.org/authors/?q=ai:ignazio.vincenzo"Reppen, A. Max"https://zbmath.org/authors/?q=ai:reppen.a-max"Soner, H. M."https://zbmath.org/authors/?q=ai:soner.halil-meteThe paper deals with a class of nonlinear integro-differential equations on a subspace of all probability measures on the real line related to the optimal control of McKean-Vlasov jump-diffusions.
The authors investigated an intrinsic notion of viscosity solutions that does not rely on the lifting to a Hilbert space and proved a comparison theorem for these solutions.Numerical method for solving a class of two-dimensional fractional optimal control problem via operational matrices of Legendre polynomialhttps://zbmath.org/1472.490552021-11-25T18:46:10.358925Z"Otaghsara, Yaser Nouralizade"https://zbmath.org/authors/?q=ai:otaghsara.yaser-nouralizade"Behroozifar, Mahmoud"https://zbmath.org/authors/?q=ai:behroozifar.mahmoud"Alipour, Mohsen"https://zbmath.org/authors/?q=ai:alipour.mohsenSummary: In this article, we present a numerical method for solving a class of two-dimensional fractional optimal control problems by the Legendre polynomial basis with fractional operational matrix. It should be mentioned that the dynamic system of the problem is based on the Caputo fractional partial derivative. This method, the dual integral is approximated by Gauss-Legendre rule, and then by using the Lagrangian equation, a nonlinear equation is obtained. This nonlinear equation set is solved by Newton's iterative method and unknown coefficients is determined. Finally, the proposed method was applied on a fractional problem with the different degree of fractional derivative. Also, the CPU time of method is exhibited. It is notable that all calculations were obtained by the Mathematica software.On a class of efficient higher order Newton-like methodshttps://zbmath.org/1472.490562021-11-25T18:46:10.358925Z"Sharma, Janak Raj"https://zbmath.org/authors/?q=ai:sharma.janak-raj"Kumar, Deepak"https://zbmath.org/authors/?q=ai:kumar.deepakSummary: Based on a two-step Newton-like scheme, we propose a three-step scheme of convergence order \(p+2 (p \geq 3)\) for solving systems of nonlinear equations. Furthermore, on the basis of this scheme a generalized \(k+2\)-step scheme with increasing convergence order \(p+2k\) is presented. Local convergence analysis including radius of convergence and uniqueness results of the methods is presented. Computational efficiency in the general form is discussed. Theoretical results are verified through numerical experimentation. Finally, the performance is demonstrated by the application of the methods on some nonlinear systems of equations.Approximation of solutions to the optimal control problems for systems with maximumhttps://zbmath.org/1472.490572021-11-25T18:46:10.358925Z"Dashkovskiy, S."https://zbmath.org/authors/?q=ai:dashkovskiy.sergey-n"Kichmarenko, O."https://zbmath.org/authors/?q=ai:kichmarenko.olga-d"Sapozhnikova, K."https://zbmath.org/authors/?q=ai:sapozhnikova.kateryna-yuThis paper describes a method for the approximate solution of optimal control problems, where the value of the input depends on the maximal values attained by the control parameter and the phase vector:
\[
\dot{x}(t) = f(t,x(t),x_m(t)) + A(x)\xi(t,u(t),u_m(t)),
\]
where \(u\) is the control parameter, and \(x_m\) and \(u_m\) represent the supremum values attained by the functions \(x\) and \(u\) in an interval \([g(t),\gamma(t)]\).
The solution proposed is to use an averaging method, namely to solve the problem
\[
\dot{y}(t) = \bar{f}(y(t),y_m(t)) + A(y(t))v(t),
\]
where \(v\) is the new control parameter and the function \(\bar{f}\) is defined as
\[
\frac{1}{T}\int_0^T f(t,x)dt.
\]
The authors propose an algorithm to find a correspondence between the averaged control \(v\) and the control \(u\), based on discretising the interval \([0,T]\) into subintervals of predefined length \(T_0\) and show that under suitable assumptions of continuity and the existence of relevant limits, there is a correspondence between the controls \(u\) and \(v\). Furthermore, the objective values of the averaged system and the original system can be made arbitrarily close.A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE systemhttps://zbmath.org/1472.490582021-11-25T18:46:10.358925Z"Holtmannspötter, Marita"https://zbmath.org/authors/?q=ai:holtmannspotter.marita"Rösch, Arnd"https://zbmath.org/authors/?q=ai:rosch.arnd"Vexler, Boris"https://zbmath.org/authors/?q=ai:vexler.borisSummary: In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of an optimal control problem governed by a simplified linear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of an elliptic PDE which has to be fulfilled at almost all times coupled with an ODE that has to hold true in almost all points in space. The state equation is discretized by a piecewise constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. For the discretization of the control we employ the same discretization technique which turns out to be equivalent to a variational discretization approach. We provide error estimates of optimal order both for the discretization of the state equation as well as for the optimal control. Numerical experiments are added to illustrate the proven rates of convergence.An iterative method for optimal control of bilateral free boundaries problemhttps://zbmath.org/1472.490592021-11-25T18:46:10.358925Z"El Yazidi, Youness"https://zbmath.org/authors/?q=ai:el-yazidi.youness"Ellabib, Abdellatif"https://zbmath.org/authors/?q=ai:ellabib.abdellatifSummary: In this paper, a bilateral free boundaries problem is considered. This kind of inverse problems appears in the theory of semiconductors and multi-phase problems. Using a shape functional and some regularization terms, an optimal control problem is formulated. In addition, we prove its solution existence. The first optimality conditions and the shape gradient are computed. With the finite element method, we write the discrete version of the optimal control problem. To design our proposed scheme, we based on the conjugate gradient, where we use the genetic algorithm to find the best initial guess for the gradient method. At each mesh regeneration, we perform a mesh refinement in order to avoid any domain singularities. Some numerical examples are shown to demonstrate the validity of the theoretical results and to prove the robustness and efficiency of the proposed scheme, especially to identify free boundaries with jump points.Bernstein-Moser-type results for nonlocal minimal graphshttps://zbmath.org/1472.490602021-11-25T18:46:10.358925Z"Cozzi, Matteo"https://zbmath.org/authors/?q=ai:cozzi.matteo"Farina, Alberto"https://zbmath.org/authors/?q=ai:farina.alberto"Lombardini, Luca"https://zbmath.org/authors/?q=ai:lombardini.lucaSummary: We prove a flatness result for entire nonlocal minimal graphs having some partial derivatives bounded from either above or below. This result generalizes fractional versions of classical theorems due to Bernstein and Moser. Our arguments rely on a general splitting result for blow-downs of nonlocal minimal graphs.
Employing similar ideas, we establish that entire nonlocal minimal graphs bounded on one side by a cone are affine.
Moreover, we show that entire graphs having constant nonlocal mean curvature are minimal, thus extending a celebrated result of Chern on classical CMC graphs.Asymptotic analysis and topological derivative for 3D quasi-linear magnetostaticshttps://zbmath.org/1472.490612021-11-25T18:46:10.358925Z"Gangl, Peter"https://zbmath.org/authors/?q=ai:gangl.peter"Sturm, Kevin"https://zbmath.org/authors/?q=ai:sturm.kevinSummary: In this paper we study the asymptotic behaviour of the quasilinear curl-curl equation of 3D magnetostatics with respect to a singular perturbation of the differential operator and prove the existence of the topological derivative using a Lagrangian approach. We follow the strategy proposed in [\textit{P. Gangl} and \textit{K. Sturm}, ESAIM, Control Optim. Calc. Var. 26, Paper No. 106, 20 p. (2020; Zbl 1459.49027)] where a systematic and concise way for the derivation of topological derivatives for quasi-linear elliptic problems in \(H^1\) is introduced. In order to prove the asymptotics for the state equation we make use of an appropriate Helmholtz decomposition. The evaluation of the topological derivative at any spatial point requires the solution of a nonlinear transmission problem. We discuss an efficient way for the numerical evaluation of the topological derivative in the whole design domain using precomputation in an offline stage. This allows us to use the topological derivative for the design optimization of an electrical machine.Nonasymptotic densities for shape reconstructionhttps://zbmath.org/1472.490622021-11-25T18:46:10.358925Z"Ibrahim, Sharif"https://zbmath.org/authors/?q=ai:ibrahim.sharif"Sonnanburg, Kevin"https://zbmath.org/authors/?q=ai:sonnanburg.kevin"Asaki, Thomas J."https://zbmath.org/authors/?q=ai:asaki.thomas-j"Vixie, Kevin R."https://zbmath.org/authors/?q=ai:vixie.kevin-rSummary: In this work, we study the problem of reconstructing shapes from simple nonasymptotic densities measured only along shape boundaries. The particular density we study is also known as the integral area invariant and corresponds to the area of a disk centered on the boundary that is also inside the shape. It is easy to show uniqueness when these densities are known for all radii in a neighborhood of \(r = 0\), but much less straightforward when we assume that we only know the area invariant and its derivatives for only one \(r > 0\). We present variations of uniqueness results for reconstruction (modulo translation and rotation) of polygons and (a dense set of) smooth curves under certain regularity conditions.On the convergence of almost minimal sets for the Hausdorff and varifold topologieshttps://zbmath.org/1472.490632021-11-25T18:46:10.358925Z"Fang, Yangqin"https://zbmath.org/authors/?q=ai:fang.yangqinThe paper under review investigates convergence properties of quasiminimal sets in \(\mathbb{R}^n\).
The class of quasiminimal sets \(\mathrm{QM}(U,M,h)\) with constant \(M\) and nondecreasing function \(h:[0,\infty[\rightarrow [0,\infty]\) is more general than the class of almost minimising sets with gauge function \(h\). It is established in Theorem~2.4 of the paper that if a sequence \(\{E_k\}\subset\mathrm{QM}(U,M_k,h_k)\) converges in the local Hausdorff distance to some set \(E\subset U\), such that the limsup of \(\{M_k\}\) is finite and the limsup of \(\{h_k\}\) has small enough value at \(0+\), and if the total Hausdorff measure of \(\{E_k\}\) tends to that of \(E\), then \(\{E_k\}\) also converges to \(E\) in the varifold sense.
In addition, it is proved in this paper that sets of positive reach in \(\mathbb{R}^n\) -- in particular, compact \(C^{1,1}\)-Euclidean submanifolds -- are almost minimal sets away from the boundary. The above results altogether yield a varifold convergence criterion for \(C^{1,1}\)-submanifolds with reaches uniformly bounded away from zero. See Corollary~3.4 in the paper under review.
Examples are provided in \S 5 to illustrate the subtleties in different modes of convergence, \emph{e.g.}, Hausdorff convergence, varifold convergence, and convergence of total mass.Degenerate free discontinuity problems and spectral inequalities in quantitative formhttps://zbmath.org/1472.490642021-11-25T18:46:10.358925Z"Bucur, Dorin"https://zbmath.org/authors/?q=ai:bucur.dorin"Giacomini, Alessandro"https://zbmath.org/authors/?q=ai:giacomini.alessandro"Nahon, Mickaël"https://zbmath.org/authors/?q=ai:nahon.mickaelAuthors' abstract: We introduce a new geometric-analytic functional that we analyse in the context of free discontinuity problems. Its main feature is that the geometric term (the length of the jump set) appears with a negative sign. This is motivated by searching quantitative inequalities for the best constants of Sobolev-Poincaré inequalities with trace terms in \(\mathbb R^n\) which correspond to fundamental eigenvalues associated to semilinear problems for the Laplace operator with Robin boundary conditions. Our method is based on the study of this new, degenerate, functional which involves an obstacle problem in interaction with the jump set. Ultimately, this becomes a mixed free discontinuity/free boundary problem occuring above/at the level of the obstacle, respectively.The area minimizing problem in conformal cones. IIhttps://zbmath.org/1472.490652021-11-25T18:46:10.358925Z"Gao, Qiang"https://zbmath.org/authors/?q=ai:gao.qiang"Zhou, Hengyu"https://zbmath.org/authors/?q=ai:zhou.hengyuIn this paper, the authors continue to study the area minimizing problem with prescribed boundary in a class of conformal cones similar to the one published by the authors in [J. Funct. Anal. 280, No. 3, Article ID 108827, 40 p. (2021; Zbl 1461.49056)]. If \(N\) is an \(n\)-dimensional open Riemannian manifold with a metric \(\sigma\), \(\mathbb{R}\) is the real line with the metric \(dr^2\), and \(\varphi(x)\) is a \(C^2\) positive function on \(N\), then \(M_\varphi=(N\times\mathbb{R},\varphi^2(x)(\sigma+dr^2))\) is called a conformal product manifold, and if \(\Omega\) is a \(C^2\) bounded domain with compact closure \(\overline\Omega\) in \(N\), then \(Q_\varphi=\Omega\times\mathbb{R}\) in \(M_\varphi\) is called a conformal cone. If \(\psi(x)\) is a \(C^1\) function on \(\partial\Omega\) and \(\Gamma\) is its graph in \(\partial\Omega\times\mathbb{R}\), then the area minimizing problem in a conformal cone \(Q_\varphi\) is to find an \(n\)-integer multiplicity current in \(\overline Q_\varphi\) to realize
\[
\min\{\mathbb{M}(T);\ T\in\mathcal{G}\ \text{and}\ \partial T=\Gamma\}, \tag{\(*\)}
\]
where \(\mathbb{M}\) is the mass of integer multiplicity currents in \(M_\varphi\), and \(\mathcal{G}\) denotes the set of \(n\)-integer multiplicity currents with compact support in \(\overline{Q}_\varphi\), that is, for any \(T\in\mathcal{G}\), its support \(\text{spt}(T)\) is contained in \(\overline\Omega\times[a,b]\) for some finite numbers \(a < b\). If \(BV(W)\) is he set of all bounded variation functions on any open set \(W\), then a key concept for the study of the problem \((*)\) is an area functional in \(BV(W)\) defined as \( \mathfrak{F}_\varphi(u,W)=\sup\left\{\int_\Omega\{\varphi^n(x)h+u\,\text{div}(\varphi^n(x)X)\}\,d\,\text{vol}\right\} \) for \(h\in C_0(W)\), \(X\in T_0(W)\), and \(h^2+\langle X,X\rangle\le 1\), where \(d\text{vol}\) and \(\text{div}\) are the volume form and the divergence of \(N\), respectively, and \(C_0(W)\) and \(T_0(W)\) denote the set of smooth functions and vector fields with compact support in \(W\), respectively. If \(u\in C^1(W)\), then \(\mathfrak{F}_\varphi(u,W)\) is the area of the graph of \(u(x)\) in \(M_\varphi\). If \(\Omega\) is the \(C^2\) domain, \(\Omega'\) is a \(C^2\) domain in \(N\) satisfying \(\Omega\subset\!\subset\Omega'\), i.e., the closure of \(\Omega\) is a compact set in \(\Omega'\), and \(\psi(x)\in C^1(\Omega'\setminus\Omega)\), then the following minimizing problem:
\[
\min\{\mathfrak{F}_\varphi(v,\Omega');\ v(x)\in BV(\Omega'), v(x)=\psi(x)\ \text{on}\ \Omega'\setminus\Omega\} \tag{\(**\)}
\]
plays an important role to solve \((*)\). If \(\Sigma\) is a minimal graph of \(u(x)\) in \(M_\varphi\) over \(\Omega\) with \(C^1\) boundary \(\psi(x)\) on \(\partial\Omega\), then the Dirichlet problem is defined as
\[
\text{div}\!\left(\frac{D u}{\sqrt{1+|Du|^2}}\right)+n\left\langle D\log\varphi,\frac{D u}{\sqrt{1+|Du|^2}}\right\rangle=0 \tag{\(*\!*\!*\)}
\]
for \(x\in\Omega\), and \(u(x)=\psi(x)\) for \(x\in\partial\Omega\), where \(\psi(x)\) is a continuous function on \(\partial\Omega\) and div is the divergence of \(\Omega\). The key idea to solve \((*)\) is to establish the connection between the problem \((*)\), the area functional minimizing problem \((**)\), and the Dirichlet problem of minimal surface equations in \(M_\varphi\). If the mean curvature \(H\) of \(\partial\Omega\) satisfies \(H_{\partial\Omega}+n\langle\vec\gamma,D\log\varphi\rangle\ge 0\) on \(\partial\Omega\), where \(\vec\gamma\) is the outward normal vector of \(\partial\Omega\) and \(H_{\partial\Omega}= \text{div}(\vec{\gamma})\), then \(\Omega\) is called \(\varphi\)-mean convex. The authors show that if \(u(x)\) is the solution to the problem \((**)\), then \(T=\partial[\![U]\!]_{\overline{Q}_\varphi}\) solves the problem \((*)\) in \(M_\varphi\), where \(U\) is the subgraph of \(u(x)\) and \([\![U]\!]\) is the corresponding integer multiplicity current. As a direct application of this result is the Dirichlet problem of minimal surface equations in \(M_\varphi\). It is shown that if \(\Omega\) is \(\varphi\)-mean convex, then the Dirichlet problem \((*\!*\!*)\) with continuous boundary data has a unique solution in \(C^2(\Omega)\cap C(\overline\Omega)\). Finally, the authors consider the existence and uniqueness of local area minimizing integer multiplicity current in \(M_\varphi\) with infinity boundary \(\Gamma\) when \(\varphi(x)\) can be written as \(\varphi(d(x,\partial N))\) which goes to \(+\infty\) as \(d(x,\partial N)\to 0\) in \(N\), where \(N\) is a compact Riemannian manifold with \(C^2\) boundary and \(d\) is the distance function in \(N\). If \(N_r=\{x\in N;\ d(x,\partial N) > r\}\), then it is shown that if there is \(r_1\) such that for any \(r\in(0,r_1)\) \(N_r\) is \(\varphi\)-mean convex, then for any \(\psi(x)\in C(\partial N)\) and \(\Gamma=(x,\psi(x))\) there is a unique local area minimizing integer multiplicity current \(T\) with infinity boundary \(\Gamma\), and \(T\) is a minimal graph in \(M_\varphi\) over \(N\).Variational problems involving unequal dimensional optimal transporthttps://zbmath.org/1472.490662021-11-25T18:46:10.358925Z"Nenna, Luca"https://zbmath.org/authors/?q=ai:nenna.luca"Pass, Brendan"https://zbmath.org/authors/?q=ai:pass.brendan-wThe paper under review studies the minimisation of functionals of the form
\[
\mathcal{J}(\mu,\nu) = \mathcal{T}_c(\mu,\nu) + \mathcal{F}(\nu) + \mathcal{G}(\mu),
\]
where $\mathcal{T}_c$ is the Monge-Kantorovich term, i.e., the optimal transport distance induced by a cost function $c$. Here $\mu$ and $\nu$ are probability measures supported in compact sets in $\mathbb{R}^m$ and $\mathbb{R}^n$, respectively. This paper is mainly concerned with the analysis of minimisers of $\mathcal{J}$ when $m>n$.
Among other results, under various assumptions on $\mathcal{F}$ and $\mathcal{G}$, a nestedness condition has been identified, which ensures that the PDE on the lower dimensional space characterising minimisers is local and degenerate elliptic. Several sufficient conditions for nestedness under suitable assumptions have been proposed. Regularity of minimisers have been discussed. Convergence of the best reply scheme has been proved. Moreover, several important examples (\emph{e.g.}, the hedonic pricing problems) have been analysed.Optimization results for the higher eigenvalues of the \(p\)-Laplacian associated with sign-changing capacitary measureshttps://zbmath.org/1472.490672021-11-25T18:46:10.358925Z"Degiovanni, Marco"https://zbmath.org/authors/?q=ai:degiovanni.marco"Mazzoleni, Dario"https://zbmath.org/authors/?q=ai:mazzoleni.darioLet \(\Omega\) be an open subset of \(\mathbb R^n\), \(|\Omega|\) be its Lebesgue measure, and \(\partial\Omega\) be its boundary. The authors consider the following boundary value problem \[ (*):\ -\text{div}(|\nabla u|^{p-2}\nabla u)=\lambda|u|^{p-2}u\text{ in }\Omega\text{ such that }u_{\restriction{_{\partial\Omega}}}=0.\] The associated eigenvalues to \((*)\) are given by: \[\lambda_{m,p}(\Omega)=\inf_{K\subset \mathcal{K}_m}\left(\sup_{u\in K}\displaystyle\int_\Omega|\nabla u(x)|^{p}dx\right)\text{ such that }m\in\mathbb N,\] \(\mathcal{K}_m=\{K\subset \mathcal{W}^{1,p}(\Omega):K\text{ is a compact and symmetric with }i(K)\ge m\},\) where \(i\) represents the Krasnosel'skii genus and \(\mathcal{W}^{1,p}(\Omega)\) is the space of functions belonging to the standard Sobolev space \(W^{1,p}(\Omega)\) and with unit \(L_p(\Omega)\)-norms. Then, the authors consider \(F\) a real-valued increasing lower semicontinuous function on \(\mathbb R^k\) and state that the following problem: \[\min\{F(\lambda_{1,p}(A),\ldots,\lambda_{k,p}(A)):A\text{ is a }p\text{-quasi open subset of }\Omega\text{ with }|A|=c\},\] has a solution where \(c\in (0,|\Omega|]\) and \(A\) is a \(p\)-quasi open set means that \(A\cup \omega_\varepsilon\) is an open set of \(\mathbb R^n\) where \(\omega_\varepsilon\) is an open subset of \(\mathbb R^n\) such that its \(p\)-capacity is less than \(\varepsilon>0\).On the unique solvability of the optimal starting control problem for the linearized equations of motion of a viscoelastic mediumhttps://zbmath.org/1472.490682021-11-25T18:46:10.358925Z"Artemov, M. A."https://zbmath.org/authors/?q=ai:artemov.mikhail-anatolevichSummary: We study an optimization problem for the linearized evolution equations of the Oldroyd model of motion of a viscoelastic medium. The equations are given in a bounded three-dimensional domain. The velocity distribution at the initial time is used as a control function. The objective functional is terminal. The existence of a unique optimal control is proved for a given set of admissible controls. A variational inequality characterizing the optimal control is derived.Optimal control of Earth pressure balance of shield tunneling machine based on dual-heuristic dynamic programminghttps://zbmath.org/1472.490692021-11-25T18:46:10.358925Z"Liu, Xuanyu"https://zbmath.org/authors/?q=ai:liu.xuanyu"Xu, Sheng"https://zbmath.org/authors/?q=ai:xu.sheng"Shao, Cheng"https://zbmath.org/authors/?q=ai:shao.chengSummary: Earth pressure balance (EPB) shield tunneling machine has been widely used in underground construction. To avoid the catastrophic accidents caused by earth pressure imbalance, the earth pressure on excavation face must be controlled balance to that in chamber. To solve this problem better, a multi-variable data-driven optimal control method for shield machine based on dual-heuristic programming (DHP) is proposed. The DHP controller is constructed with action network, model network, and critic network based on back-propagation neural networks (BPNNs). Following Bellman's principle of optimality, a cost function of DHP controller for the chamber's earth pressure is presented, which simplifies a multi-level optimization to a single-level optimization. To minimize the cost function, the action network utilizes the critic network's error to achieve multi-variable optimization, and the optimal control parameters for the tunneling process are obtained at last. The simulation results show that the method can effectively control the earth pressure balance. Even in case of disturbance, the system has strong anti-interference ability and the control process is also quicker and steadier.\(\mathbb{B}\)-spaces are KKM spaceshttps://zbmath.org/1472.520032021-11-25T18:46:10.358925Z"Park, Sehie"https://zbmath.org/authors/?q=ai:park.sehieSummary: A subset \(B\) of \(\mathbb{R}_+^n\) is \(\mathbb{B}\)-convex if for all \(x_1,x_2\in B\) and all \(t\in[0,1]\) one has \(tx_1\vee x_2\in B\). These sets were first investigated in [\textit{W. Briec} and \textit{C. Horvath}, Optimization 53, No. 2, 103--127 (2004; Zbl 1144.90506)]. In this paper, we show that any finite dimensional \(\mathbb{B}\)-space is a KKM space, that is, a space satisfying the abstract form of the celebrated Knaster-Kuratowski-Mazurkiewicz theorem appeared in 1929 and its open-valued version. Therefore, a \(\mathbb{B}\)-space satisfies a large number of the KKM theoretic results appeared in the literature.Leaves decompositions in Euclidean spaceshttps://zbmath.org/1472.520052021-11-25T18:46:10.358925Z"Ciosmak, Krzysztof J."https://zbmath.org/authors/?q=ai:ciosmak.krzysztof-jSummary: We partly extend the localisation technique from convex geometry to the multiple constraints setting.
For a given 1-Lipschitz map \(u:\mathbb{R}^n\to\mathbb{R}^m\), \(m\leq n\), we define and prove the existence of a partition of \(\mathbb{R}^n\), up to a set of Lebesgue measure zero, into maximal closed convex sets such that restriction of \(u\) is an isometry on these sets.
We consider a disintegration, with respect to this partition, of a log-concave measure. We prove that for almost every set of the partition of dimension \(m\), the associated conditional measure is log-concave. This result is proven also in the context of the curvature-dimension condition for weighted Riemannian manifolds. This partially confirms a conjecture of Klartag.On the stability for Alexandrov's soap bubble theoremhttps://zbmath.org/1472.530132021-11-25T18:46:10.358925Z"Magnanini, Rolando"https://zbmath.org/authors/?q=ai:magnanini.rolando"Poggesi, Giorgio"https://zbmath.org/authors/?q=ai:poggesi.giorgioSummary: Alexandrov's Soap Bubble Theorem dates back to 1958 and states that a compact embedded hypersurface in \(\mathbb{R}^N\) with constant mean curvature must be a sphere. For its proof, A. D. Alexandrov invented his reflection principle. In [Indiana Univ. Math. J. 26, 459--472 (1977; Zbl 0391.53019)], \textit{R. C. Reilly} gave an alternative proof, based on integral identities and inequalities, connected with the torsional rigidity of a bar. In this article we study the stability of the spherical symmetry: the question is how near is a hypersurface to a sphere, when its mean curvature is near to a constant in some norm?
We present a stability estimate that states that a compact hypersurface \(\Gamma \subset \mathbb{R}^N\) can be contained in a spherical annulus whose interior and exterior radii, say \(\rho_i\) and \(\rho_e\), satisfy the inequality \[\rho_e - \rho_i \leq C \| H - H_0\|_{L^1(\Gamma)}^{\tau_N},\] where \(\tau_N = 1/2\) if \(N = 2, 3\), and \(\tau_N = 1/(N + 2)\) if \(N \geq 4\). Here, \(H\) is the mean curvature of \(\Gamma,\ H_0\) is some reference constant, and \(C\) is a constant that depends on some geometrical and spectral parameters associated with \(\Gamma\). This estimate improves previous results in the literature under various aspects. We also present similar estimates for some related overdetermined problems.Embedding surfaces inside small domains with minimal distortionhttps://zbmath.org/1472.530532021-11-25T18:46:10.358925Z"Shachar, Asaf"https://zbmath.org/authors/?q=ai:shachar.asafSummary: Given two-dimensional Riemannian manifolds \(\mathcal{M},\mathcal{N} \), we prove a lower bound on the distortion of embeddings \(\mathcal{M}\rightarrow \mathcal{N} \), in terms of the areas' discrepancy \(V_{\mathcal{N}}/V_{\mathcal{M}} \), for a certain class of distortion functionals. For \(V_{\mathcal{N}}/V_{\mathcal{M}} \ge 1/4\), homotheties, provided they exist, are the unique energy minimizing maps attaining the bound, while for \(V_{\mathcal{N}}/V_{\mathcal{M}} \le 1/4\), there are non-homothetic minimizers. We characterize the maps attaining the bound, and construct explicit non-homothetic minimizers between disks. We then prove stability results for the two regimes. We end by analyzing other families of distortion functionals. In particular we characterize a family of functionals where no phase transition in the minimizers occurs; homotheties are the energy minimizers for all values of \(V_{\mathcal{N}}/V_{\mathcal{M}} \), provided they exist.Correction to: ``The total intrinsic curvature of curves in Riemannian surfaces''https://zbmath.org/1472.530632021-11-25T18:46:10.358925Z"Mucci, Domenico"https://zbmath.org/authors/?q=ai:mucci.domenico"Saracco, Alberto"https://zbmath.org/authors/?q=ai:saracco.albertoFrom the text: In the authors' paper [ibid. 70, No. 1, 521--557 (2021; Zbl 1466.53064)], in the statements of the main results, Theorems 1--9 and Proposition 3, one has to assume in addition that the curve \(\mathbf{c}\) is rectifiable.A Björling representation for Jacobi fields on minimal surfaces and soap film instabilitieshttps://zbmath.org/1472.530662021-11-25T18:46:10.358925Z"Alexander, Gareth P."https://zbmath.org/authors/?q=ai:alexander.gareth-p"Machon, Thomas"https://zbmath.org/authors/?q=ai:machon.thomasSummary: We develop a general framework for the description of instabilities on soap films using the Björling representation of minimal surfaces. The construction is naturally geometric and the instability has the interpretation as being specified by its amplitude and transverse gradient along any curve lying in the minimal surface. When the amplitude vanishes, the curve forms part of the boundary to a critically stable domain, while when the gradient vanishes the Jacobi field is maximal along the curve. In the latter case, we show that the Jacobi field is maximally localized if its amplitude is taken to be the lowest eigenfunction of a one-dimensional Schrödinger operator. We present examples for the helicoid, catenoid, circular helicoids and planar Enneper minimal surfaces, and emphasize that the geometric nature of the Björling representation allows direct connection with instabilities observed in soap films.Extending the realm of Horvath spaceshttps://zbmath.org/1472.540112021-11-25T18:46:10.358925Z"Park, Sehie"https://zbmath.org/authors/?q=ai:park.sehieSummary: A KKM space is an abstract convex space satisfying the abstract form of the KKM theorem and its open-valued version. In this article we introduce a typical subclass of KKM spaces called the Horvath spaces including \(c\)-spaces due to Horvath. We show that hyperbolic metric spaces, metric spaces with continuous midpoints, metric spaces with global nonpositive curvature (NPC) and convex hull finite property (CHFP), certain Riemannian manifolds, and \(\mathbb{B}\)-spaces are relatively new examples of Horvath spaces. Many of their properties are introduced by following our previous works [the author, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 12, 4352--4364 (2008; Zbl 1163.47044); Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 4, 1028--1042 (2010; Zbl 1214.47042)].Contraction principle for trajectories of random walks and Cramér's theorem for kernel-weighted sumshttps://zbmath.org/1472.600532021-11-25T18:46:10.358925Z"Vysotsky, Vladislav"https://zbmath.org/authors/?q=ai:vysotsky.vladislav-vSummary: In 2013 \textit{A. A. Borovkov} and \textit{A. A. Mogulskii} [Theory Probab. Appl. 57, No. 1, 1--27 (2013; Zbl 1279.60037); translation from Teor. Veroyatn. Primen. 57, No. 1, 3--34 (2012)] proved a weaker-than-standard ``metric'' large deviations principle (LDP) for trajectories of random walks in \(\mathbb{R}^d\) whose increments have the Laplace transform finite in a neighbourhood of zero. We prove that general metric LDPs are preserved under uniformly continuous mappings. This allows us to transform the result of Borovkov and Mogulskii into standard LDPs. We also give an explicit integral representation of the rate function they found. As an application, we extend the classical Cramér theorem by proving an LPD for kernel-weighted sums of i.i.d. random vectors in \(\mathbb{R}^d\).A mean field games approach to cluster analysishttps://zbmath.org/1472.620892021-11-25T18:46:10.358925Z"Aquilanti, Laura"https://zbmath.org/authors/?q=ai:aquilanti.laura"Cacace, Simone"https://zbmath.org/authors/?q=ai:cacace.simone"Camilli, Fabio"https://zbmath.org/authors/?q=ai:camilli.fabio"De Maio, Raul"https://zbmath.org/authors/?q=ai:de-maio.raulSummary: In this paper, we develop a Mean Field Games approach to Cluster Analysis. We consider a finite mixture model, given by a convex combination of probability density functions, to describe the given data set. We interpret a data point as an agent of one of the populations represented by the components of the mixture model, and we introduce a corresponding optimal control problem. In this way, we obtain a multi-population Mean Field Games system which characterizes the parameters of the finite mixture model. Our method can be interpreted as a continuous version of the classical Expectation-Maximization algorithm.Convergence of the forward-backward method for split null-point problems beyond cocoercivenesshttps://zbmath.org/1472.650752021-11-25T18:46:10.358925Z"Moudafi, Abdellatif"https://zbmath.org/authors/?q=ai:moudafi.abdellatif"Shehu, Yekini"https://zbmath.org/authors/?q=ai:shehu.yekiniSummary: The forward-backward algorithm is one of the most attractive algorithm for finding zeroes of the sum of two maximal monotone operators, with one being single-valued. However, it requires the single-valued part to be co-coercive, thus precluding its use in many applications. The aim of this paper is to present and investigate the asymptotic behavior of a forward-backward algorithm with Bregman distances for solving constrained split null-point problems beyond co-coerciveness of its single-valued part. Special attention is given to constrained composite minimization and we illustrate the potential of this approach by answering a question by \textit{H.-K. Xu} [Linear Nonlinear Anal. 4, No. 1, 135--144 (2018; Zbl 1458.94147)] on the non applicability of the proximal gradient algorithm when the \(l_p\)-norm is used to measure the errors in signal processing.Efficient modified techniques of invariant energy quadratization approach for gradient flowshttps://zbmath.org/1472.651002021-11-25T18:46:10.358925Z"Liu, Zhengguang"https://zbmath.org/authors/?q=ai:liu.zhengguang"Li, Xiaoli"https://zbmath.org/authors/?q=ai:li.xiaoliSummary: Two novel and efficient modified techniques based on recently developed stabilized invariant energy quadratization (IEQ) approach to deal with nonlinear terms in gradient flows are proposed in this paper. We proved the unconditional energy stability for a class of gradient flows and their semi-discrete schemes carefully and rigorously. One of the contributions for this approach is that we succeeded in finding suitable positive preserving functions in square root and do not need to add a positive constant which cannot be fixed before computing. Secondly, all nonlinear terms can be treated semi-explicitly, and one only needs to solve three decoupled linear equations with constant coefficients at each time step. Finally, several numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.The hessian Riemannian flow and Newton's method for effective Hamiltonians and Mather measureshttps://zbmath.org/1472.651102021-11-25T18:46:10.358925Z"Gomes, Diogo A."https://zbmath.org/authors/?q=ai:gomes.diogo-luis-aguiar"Yang, Xianjin"https://zbmath.org/authors/?q=ai:yang.xianjinThe authors suggest two methods, the Hessian Riemannian flow and Newton's method, to calculate simultaneously the effective Hamiltonian and the Mather measures. The convergence of the Hessian Riemannian flow in the continuous setting is proved. For the discrete case, the existence and the convergence of the Hessian Riemannian flow are shown. A variant of Newton's method, which improves the performance of the Hessian Riemannian flow, is used. Some numerical tests are presented to show that the algorithms preserve the non-negativity of Mather measures and are more stable than related methods in problems that are close to singular.Efficient computational approach for generalized fractional KdV-Burgers equationhttps://zbmath.org/1472.651312021-11-25T18:46:10.358925Z"Rida, Saad Z."https://zbmath.org/authors/?q=ai:rida.saad-zagloul"Hussien, Hussien S."https://zbmath.org/authors/?q=ai:hussien.hussien-shafeiSummary: A collocation method based on double summations of Mittag-Leffler functions is proposed to solve the Korteweg-de Vries (KdV) and Burgers equation of fractional order with initial-boundary conditions. The resulting algebraic system is constructed as a constrained optimization problem and optimized to obtain the unknown coefficients. Error analysis of the approximation solution is studied. Simulations of the results are studied graphically through representations for the effect of fractional order parameters and time levels. The results ensure that the proposed method is accurate and efficient.Machine learning from a continuous viewpoint. Ihttps://zbmath.org/1472.681362021-11-25T18:46:10.358925Z"E, Weinan"https://zbmath.org/authors/?q=ai:e.weinan"Ma, Chao"https://zbmath.org/authors/?q=ai:ma.chao"Wu, Lei"https://zbmath.org/authors/?q=ai:wu.lei.1|wu.lei.2|wu.lei.3|wu.lei.4Summary: We present a continuous formulation of machine learning, as a problem in the calculus of variations and differential-integral equations, in the spirit of classical numerical analysis. We demonstrate that conventional machine learning models and algorithms, such as the random feature model, the two-layer neural network model and the residual neural network model, can all be recovered (in a scaled form) as particular discretizations of different continuous formulations. We also present examples of new models, such as the flow-based random feature model, and new algorithms, such as the smoothed particle method and spectral method, that arise naturally from this continuous formulation. We discuss how the issues of generalization error and implicit regularization can be studied under this framework.Control of chaotic systems by deep reinforcement learninghttps://zbmath.org/1472.681712021-11-25T18:46:10.358925Z"Bucci, M. A."https://zbmath.org/authors/?q=ai:bucci.michele-alessandro"Semeraro, O."https://zbmath.org/authors/?q=ai:semeraro.onofrio"Allauzen, A."https://zbmath.org/authors/?q=ai:allauzen.alexandre"Wisniewski, G."https://zbmath.org/authors/?q=ai:wisniewski.grzegorz"Cordier, L."https://zbmath.org/authors/?q=ai:cordier.laurent"Mathelin, L."https://zbmath.org/authors/?q=ai:mathelin.lionelSummary: Deep reinforcement learning (DRL) is applied to control a nonlinear, chaotic system governed by the one-dimensional Kuramoto-Sivashinsky (KS) equation. DRL uses reinforcement learning principles for the determination of optimal control solutions and deep neural networks for approximating the value function and the control policy. Recent applications have shown that DRL may achieve superhuman performance in complex cognitive tasks. In this work, we show that using restricted localized actuation, partial knowledge of the state based on limited sensor measurements and model-free DRL controllers, it is possible to stabilize the dynamics of the KS system around its unstable fixed solutions, here considered as target states. The robustness of the controllers is tested by considering several trajectories in the phase space emanating from different initial conditions; we show that DRL is always capable of driving and stabilizing the dynamics around target states. The possibility of controlling the KS system in the chaotic regime by using a DRL strategy solely relying on local measurements suggests the extension of the application of RL methods to the control of more complex systems such as drag reduction in bluff-body wakes or the enhancement/diminution of turbulent mixing.A unified approach for topology optimization with local stress constraints considering various failure criteria: von Mises, Drucker-Prager, Tresca, Mohr-Coulomb, Bresler-Pister and Willam-Warnkehttps://zbmath.org/1472.741792021-11-25T18:46:10.358925Z"Giraldo-Londoño, Oliver"https://zbmath.org/authors/?q=ai:giraldo-londono.oliver"Paulino, Glaucio H."https://zbmath.org/authors/?q=ai:paulino.glaucio-hSummary: An interesting, yet challenging problem in topology optimization consists of finding the lightest structure that is able to withstand a given set of applied loads without experiencing local material failure. Most studies consider material failure via the von Mises criterion, which is designed for ductile materials. To extend the range of applications to structures made of a variety of different materials, we introduce a unified yield function that is able to represent several classical failure criteria including von Mises, Drucker-Prager, Tresca, Mohr-Coulomb, Bresler-Pister and Willam-Warnke, and use it to solve topology optimization problems with local stress constraints. The unified yield function not only represents the classical criteria, but also provides a smooth representation of the Tresca and the Mohr-Coulomb criteria -- an attribute that is desired when using gradient-based optimization algorithms. The present framework has been built so that it can be extended to failure criteria other than the ones addressed in this investigation. We present numerical examples to illustrate how the unified yield function can be used to obtain different designs, under prescribed loading or design-dependent loading (e.g. self-weight), depending on the chosen failure criterion.A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalitieshttps://zbmath.org/1472.810902021-11-25T18:46:10.358925Z"Antunes, Pedro R. S."https://zbmath.org/authors/?q=ai:antunes.pedro-ricardo-simao"Benguria, Rafael D."https://zbmath.org/authors/?q=ai:benguria.rafael-d"Lotoreichik, Vladimir"https://zbmath.org/authors/?q=ai:lotoreichik.vladimir"Ourmières-Bonafos, Thomas"https://zbmath.org/authors/?q=ai:ourmieres-bonafos.thomaslet \(\Omega \subset {\mathbb R}^2\) be a \(C^\infty\) simply connected domain and let \(n = (n_1,n_2)^\top\) be the outward pointing normal field on \(\partial\Omega\). The Dirac operator with infinite mass boundary conditions in \(L^2(\Omega,{\mathbb C}^2)\) is defined as \[D^\Omega := \begin{pmatrix} 0 & -2\mathrm{i}\partial_z\\
-2\mathrm{i}\partial_{\bar z} & 0 \end{pmatrix}, \] with domain \(\{ u = (u_1,u_2)^\top \in H^1(\Omega,{\mathbb C}^2) : u_2 = \mathrm{i} \mathbf{n}u_1 \text{ on }\partial\Omega \},\) where \(\mathbf{n} := n_1 + \mathrm{i} n_2\) and \(\partial_z, \partial_{\bar{z}}\) are the Wirtinger operators. The spectrum of \(D^\Omega\) is symmetric with respect to the origin and constituted of eigenvalues of finite multiplicity \[ \cdots \leq -E_k(\Omega) \leq\cdots \leq-E_{1}(\Omega) < 0 < E_{1}(\Omega) \leq \cdots \leq E_k(\Omega) \leq \cdots.\] The authors prove the following estimate \[E_1(\Omega) \leq \frac{|\partial\Omega|}{(\pi r_i^2 + |\Omega|)}E_1({\mathbb D}) \] with equality if and only if \(\Omega\) is a disk, where \(r_i\) is the inradius of \(\Omega\) and \(\mathbb D\) is the unit disk. \par The second main result of this paper is the following non-linear variational characterization of \(E_1(\Omega)\). \(E > 0\) is the first non-negative eigenvalue of \(D^\Omega\) if and only if \(\mu^\Omega(E) = 0\), where \[\mu^\Omega(E) := \inf\limits_{u} \frac{4 \int_\Omega |\partial_{\bar z} u|^2 dx - E^2 \int_{\Omega}|u|^2dx + E \int_{\partial\Omega} |u|^2 ds}{\int_\Omega |u|^2 dx}.\] \par The authors propose the following conjecture \[\mu^\Omega(E) \geq \frac{\pi}{|\Omega|}\mu^{\mathbb D}\Big(\sqrt{\frac{|\Omega|}{\pi}}E\Big), \forall E>0\] and provide numerical evidences supporting it. This conjecture implies the validity of the Faber-Krahn-type inequality \(E_1(\Omega) \geq \sqrt{\frac{\pi}{|\Omega|}} E_1({\mathbb D})\) (it is still an open question).A neural network-based policy iteration algorithm with global \(H^2\)-superlinear convergence for stochastic games on domainshttps://zbmath.org/1472.820302021-11-25T18:46:10.358925Z"Ito, Kazufumi"https://zbmath.org/authors/?q=ai:ito.kazufumi"Reisinger, Christoph"https://zbmath.org/authors/?q=ai:reisinger.christoph"Zhang, Yufei"https://zbmath.org/authors/?q=ai:zhang.yufeiThe following Hamilton-Jacobi-Bellman-Isaacs (HJBI) nonhomogeneous Dirichlet boundary value problem is considered: $F(u): =-a^{ij}(x)\partial_{ij}u+ G(x,u,\nabla u)=0$, for a.e. $x\in \Omega$, $\tau u=g$, on $\partial\Omega$, with a nonlinear Hamiltonian, $G(x,u,\nabla u)=\max_{\alpha \in A}\min_{\beta \in B}(b^i(x,\alpha,\beta)$ $\partial_iu(x)+c(x,\alpha,\beta)u(x) -f(x,\alpha,\beta))$. The aim here is to investigate some numerical algorithms for solving this kind of problems. The second section is devoted to basics. Under some assumptions on the coefficients, the uniqueness of the strong solution in $H^2(\Omega)$ is proved. In the third section one presents the policy iteration algorithm -- Algorithm 1 -- for the Dirichlet problem, followed by the convergence analysis. Results on semi smoothness of the HJBI operator, q-superlinear convergence of Algorithm 1 and global convergence of Algorithm 1 are proved. In the fourth section the authors develop an inexact policy algorithm for the stated Dirichlet problem. The idea is to compute an approximate solution for the linear Dirichlet problem for the iteration $u^{k+1}\in H^2(\Omega)$ in Algorithm 1, by solving an optimization problem over a set of trial functions, within a given accuracy. The new inexact policy iteration algorithm for the Dirichlet problem -- Algorithm 2 -- is presented and under some special assumptions a result on global superlinear convergence is proved. In the fifth section we find an extension of the developed iteration scheme to other boundary value problems and a connection to the artificial neural network technology. One considers a HJBI oblique derivative problem
\[
F(u): =-a^{ij}(x)\partial _{ij}u+G(x,u,\nabla u)=0,\text{ for a.e. }x\in\Omega,
\]
$Bu:=\gamma^i\tau(\partial_iu)+\gamma^0$ $\tau u-g$, on $\partial\Gamma$. Under some assumptions on the coefficients, one proves that the oblique derivative problem admits a unique strong solution in $H^2(\Omega)$. For solving the oblique derivative problem one develops a neural network-based policy iteration algorithm, Algorithm 3. The global superlinear convergence of Algorithm 3 is proved. In the sixth section, there is a large discussion on applications of the developed algorithms to the stochastic Zermelo navigation problem. Some fundamental results used in the article are resumed at the end of the paper.Optimal control model of immunotherapy for autoimmune diseaseshttps://zbmath.org/1472.820312021-11-25T18:46:10.358925Z"Costa, M. Fernanda P."https://zbmath.org/authors/?q=ai:costa.m-fernanda-p"Ramos, M. P."https://zbmath.org/authors/?q=ai:ramos.m-p-machado"Ribeiro, C."https://zbmath.org/authors/?q=ai:ribeiro.cassio-b|ribeiro.conceicao|ribeiro.claudio-d|ribeiro.cristina|ribeiro.c-c-h|ribeiro.carlos-h-c|ribeiro.celso-carneiro|ribeiro.cassilda|ribeiro.carlos-f-m|ribeiro.clovis-a|ribeiro.carlos-augusto-david|ribeiro.claudia"Soares, A. J."https://zbmath.org/authors/?q=ai:soares.ana-jacintaSummary: In this work, we develop a new mathematical model to evaluate the impact of drug therapies on autoimmunity disease. We describe the immune system interactions at the cellular level, using the kinetic theory approach, by considering self-antigen presenting cells, self-reactive T cells, immunosuppressive cells, and Interleukin-2 (IL-2) cytokines. The drug therapy consists of an intake of Interleukin-2 cytokines which boosts the effect of immunosuppressive cells on the autoimmune reaction. We also derive the macroscopic model relative to the kinetic system and study the wellposedness of the Cauchy problem for the corresponding system of equations. We formulate an optimal control problem relative to the model so that the quantity of both the self-reactive T cells that are produced in the body and the Interleukin-2 cytokines that are administrated is simultaneously minimized. Moreover, we perform some numerical tests in view of investigating optimal treatment strategies and the results reveal that the optimal control approach provides good-quality approximate solutions and shows to be a valuable procedure in identifying optimal treatment strategies.Corrigendum to: ``Optimal policies for a finite-horizon batching inventory model''https://zbmath.org/1472.900062021-11-25T18:46:10.358925Z"Al-Khamis, T. M."https://zbmath.org/authors/?q=ai:alkhamis.talal-m"Benkherouf, L."https://zbmath.org/authors/?q=ai:benkherouf.lakdere"Omar, M."https://zbmath.org/authors/?q=ai:omar.mohd|omar.m-rafiq|omar.m-hafidz|omar.mohamed-a|omar.mohamed-kamal|omar.mawloud|omar.mohamed-khaledFrom the text: The authors of [ibid. 45, No. 10, 2196--2202 (2014; Zbl 1317.90011)] advise that Example 4.1 on page 2201 contains some errors. The set-up cost \(K\) should be 800 and
the last column \(C_n\) should be 4546 for \(b=0.1\) and 13191 for \(b=-0.1\).Robust Farkas-Minkowski constraint qualification for convex inequality system under data uncertaintyhttps://zbmath.org/1472.900782021-11-25T18:46:10.358925Z"Li, Xiao-Bing"https://zbmath.org/authors/?q=ai:li.xiaobing"Al-Homidan, Suliman"https://zbmath.org/authors/?q=ai:al-homidan.suliman-s"Ansari, Qamrul Hasan"https://zbmath.org/authors/?q=ai:ansari.qamrul-hasan"Yao, Jen-Chih"https://zbmath.org/authors/?q=ai:yao.jen-chihThe authors consider the nonempty assumed solution set \(S=\{x\in \mathbb{R}^n \mid g_i(x,u_i)\le 0\; \forall\,u_i\in \mathcal{U}_i,\, i\in \mathcal{I}=1,2,\dots k \}\) of a robust finite inequality system with compact convex uncertainty sets \(\mathcal{U}_i\) and convex-concave finite valued functions \(g_i:\mathbb{R}^n\times\mathbb{R}^m\rightarrow\mathbb{R}\), \(i\in \mathcal{I}\). They show (Th. 1) that the robust global error bound (RGEB) \(\alpha \inf_{y\in S}\|x-y\|\le \sum_{i\in \mathcal{I}}[\sup_{u_i\in \mathcal{U}_i}g_i(x,u_i)]_+\) on \(\mathbb{R}^n\) for some \(\alpha>0\) is sufficient for the validity of the robust Farkas-Minkowski constraint qualification (FMCQ) according to \(\mathrm{epi}\,\delta^*_S=\bigcup_{\lambda_i>0,\,\,u_i\in \mathcal{U}_i,\,i\in \mathcal{I}}\mathrm{epi}\,(\sum_{i\in \mathcal{I}}\lambda_ig_i(x,u_i))^*\). \(\mathrm{epi} f\) is the epigraph of an extended real-valued function \(f\) and \(\delta^*_S\) denotes the support functional of the convex set \(S\). Hence the union is a closed convex cone. Example 3.2 demonstrates that the concavity of \(g_i\) w.r.t. \(u_i\) is essential. Conditions (FMCQ) and (REGB) are equivalent for the special case of in \(x\) positively semidefinite quadratic forms \(g_i\) with coefficients \(u_i\) belonging to a scenario uncertainty set. In the last three rows of the proof of Th. 1, the subset sign should be replaced by the corresponding superset sign for getting equality.Efficient algorithms for distributionally robust stochastic optimization with discrete scenario supporthttps://zbmath.org/1472.900802021-11-25T18:46:10.358925Z"Zhang, Zhe"https://zbmath.org/authors/?q=ai:zhang.zhe"Ahmed, Shabbir"https://zbmath.org/authors/?q=ai:ahmed.shabbir.1|ahmed.shabbir"Lan, Guanghui"https://zbmath.org/authors/?q=ai:lan.guanghuiStrong duality in minimizing a quadratic form subject to two homogeneous quadratic inequalities over the unit spherehttps://zbmath.org/1472.900822021-11-25T18:46:10.358925Z"Nguyen, Van-Bong"https://zbmath.org/authors/?q=ai:nguyen.van-bong"Nguyen, Thi Ngan"https://zbmath.org/authors/?q=ai:nguyen.thi-ngan"Sheu, Ruey-Lin"https://zbmath.org/authors/?q=ai:sheu.ruey-linThe authors study the strong duality for an optimization problem to minimize a homogeneous quadratic function subject to two homogeneous quadratic constraints over the unit sphere, called Problem (P) in this paper. When a feasible (P) fails to have a Slater point, they show that (P) always adopts the strong duality. When (P) has a Slater point, the authors propose a set of conditions, called ``Property J'', on an SDP relaxation of (P) and its conical dual. They show that (P) has the strong duality if and only if there exists at least one optimal solution to the SDP relaxation of (P) which fails Property J. The used techniques are based on various extensions of the \(S\)-lemma as well as the matrix rank-one decomposition procedure introduced by \textit{W. Ai} and \textit{S. Zhang} [SIAM J. Optim. 19, No. 4, 1735--1756 (2009; Zbl 1187.90290)]. Many nontrivial examples are constructed to help understand the mechanism.Inexact descent methods with convex objective functions in Banach spaceshttps://zbmath.org/1472.900922021-11-25T18:46:10.358925Z"Reich, Simeon"https://zbmath.org/authors/?q=ai:reich.simeon"Zaslavski, Alexander J."https://zbmath.org/authors/?q=ai:zaslavski.alexander-jSummary: Given a Lipschitz and convex objective function on a Banach space, we revisit the class of regular vector fields introduced in our previous work on descent methods. We study the behavior of the values of the objective function for a certain inexact process generated by a regular vector field and show that the values of the objective function converge to its infimum.Optimality conditions for nonconvex constrained optimization problemshttps://zbmath.org/1472.901002021-11-25T18:46:10.358925Z"Mashkoorzadeh, F."https://zbmath.org/authors/?q=ai:mashkoorzadeh.feryal"Movahedian, N."https://zbmath.org/authors/?q=ai:movahedian.nooshin"Nobakhtian, S."https://zbmath.org/authors/?q=ai:nobakhtian.soghraThe authors consider nonconvex mathematical programming problems with a tangentially convex objective \(f:\mathbb{R}^n \rightarrow \mathbb{R}\cup \{\infty\}\) and tangentially convex and continuous inequality constraints \(g_i(x)\le 0\), \(i=1,2,\dots,m<\infty\), defining the feasible set \(S\). \(f\) is called tangentially convex at \(\bar x \in \mathrm{dom }f\) [\textit{C. Lemaréchal}, Optimization 17, 827--858 (1986; Zbl 0613.49017)] whenever the directional derivative \(f'(\bar x,v)\) of \(f\) exists and is finite for all directions \(v\in \mathbb{R}^n\) and the function \(v\mapsto f'(\bar x,v)\) is convex on \(\mathbb{R}^n\). The main role playes its tangential subdifferential \(\partial f(\bar x):=\partial f'(\bar x,0)\) already considered in [\textit{A. D. Ioffe} and \textit{V. M. Tikhomirov}, Theorie der Extremalprobleme. (Teorija ekstremal'nyh zadac.) (Russian). Nelineinyi analiz i ego prilozenija. Moskau: Verlag 'Nauka', Hauptredaktion für physikalisch-mathematische Literatur (1974; Zbl 0292.90042), Chap. 4.4; \textit{B. N. Pshenichnyj}, Sov. Math., Dokl. 10, 70--72 (1969; Zbl 0227.90062); translation from Dokl. Akad. Nauk SSSR 184, 285--287 (1969); Necessary conditions for an extremum. New York, NY: Marcel Dekker, Inc. (1971; Zbl 0212.23902)]. First the standard regularity conditions extended to tangential subdifferential notation are shown to satisfy similar relations to each other as known from the differential case. Non valid relations are verified by counter examples. If \(f\) is Lipschitzian near \(\bar x\), then necessary local optimality conditions are shown like KKT-rule under regularity and without regularity \(f^\prime(\bar x,v)\le 0\) on the tangent cone being locally sufficient in case of \(f^\prime(\bar x,v)> 0\) on the tangent cone. A KKT condition at \(\bar x\) is sufficient for global optimality under Slater's condition and pseudoconvexity of \(f\). All optimality conditions are illustrated by examples. A handicap for reading the paper are several misprints in definitions corrected in a just now published more general paper on DTC programming \url{https://doi.org/10.1007/s11750-021-00615-z}.Optimality conditions of a nonsmooth composite minimization problemhttps://zbmath.org/1472.901262021-11-25T18:46:10.358925Z"Atarzadeh, Sahar"https://zbmath.org/authors/?q=ai:atarzadeh.sahar"Fakhar, Majid"https://zbmath.org/authors/?q=ai:fakhar.majid"Zafarani, Jafar"https://zbmath.org/authors/?q=ai:zafarani.jafarSummary: Paper considered a nonsmooth composite minimization problem with inequality constraints ((NCMP) in short). For this problem and in the setting of reflexive Banach spaces, the Fritz John (FJ) optimality condition for this problem at points which are not necessarily local minima is obtained. By using this result, the FJ optimality condition for a nonsmooth optimization problem with inequality constraints is established. Moreover, by applying the Clarke, the Michel-Penot and modified upper Dini derivatives, some equivalent conditions for the Karush-Kuhn-Tucker (KKT) optimality condition of (NCMP) without using the FJ optimality conditions are deduced.Avoiding critical multipliers and slow convergence of primal-dual methods for fully stable minimizershttps://zbmath.org/1472.901342021-11-25T18:46:10.358925Z"Mordukhovich, Boris S."https://zbmath.org/authors/?q=ai:mordukhovich.boris-sSummary: This paper discusses the current stage of the author's conjecture on relationships between criticality of Lagrange multipliers in various classes of constrained optimization problems and full stability of local minimizers. It has been realized that the existence of critical multipliers associated with a given local minimizer leads to slow convergence of major primal-dual algorithms to calculate such minimizers. The paper demonstrates that for large classes of constrained optimization problems critical multipliers do not appear if the local minimizer in question is fully stable. The developed approach is mainly based on advanced tools of second-order variational analysis and generalized differentiation.On the existence of solutions and Tikhonov regularization of hemivariational inequality problemshttps://zbmath.org/1472.901442021-11-25T18:46:10.358925Z"Tang, Guo-ji"https://zbmath.org/authors/?q=ai:tang.guoji"Wan, Zhongping"https://zbmath.org/authors/?q=ai:wan.zhongping"Wang, Xianfu"https://zbmath.org/authors/?q=ai:wang.xianfu.1|wang.xianfuIn this paper, the authors establish a Tikhonov regularization theory for a class of hemivariational inequalities. To this end, they prove a very general existence result for the class of hemeivariational inequalities provided that the mapping has the so-called hemivariational inequality property and satisfies a rather weak coercivity condition. Based on the existence result, they derive the Tikhonov regularization result.Dynamic programming and optimal control for vector-valued functions: a state-of-the-art reviewhttps://zbmath.org/1472.901482021-11-25T18:46:10.358925Z"Abdelaziz, Fouad Ben"https://zbmath.org/authors/?q=ai:ben-abdelaziz.fouad"La Torre, Davide"https://zbmath.org/authors/?q=ai:la-torre.davide"Alaya, Houda"https://zbmath.org/authors/?q=ai:alaya.houdaSummary: This paper aims to present a state-of-the-art review of recent development within the areas of dynamic programming and optimal control for vector-valued functions.Optimal consumption and R\&D investment for a risk-averse entrepreneurhttps://zbmath.org/1472.910152021-11-25T18:46:10.358925Z"Wang, Ming-Hui"https://zbmath.org/authors/?q=ai:wang.minghui"Huang, Nan-Jing"https://zbmath.org/authors/?q=ai:huang.nan-jingSummary: We consider an optimal consumption problem for a risk-averse entrepreneur who invests into a research and development (R\&D) project and intends to choose an optimal timing to invest into an other project. We put the decision making problem into a real option game framework. We divide the problem into two stages: the pre-investment and post-investment problems. For the first problem, the entrepreneur puts his money into an R\&D project whilst choosing a suitable timing to invest into a project. For the second problem, the entrepreneur just needs to control personal consumption and the investment rate to obtain personal maximum utility. By the method of dynamic programming, we can change the pre-investment and post-investment problems into a nonlinear variational inequality and a nonlinear partial differential equation, respectively. Then, the value function, the optimal investment rate and the optimal consumption of the entrepreneur are given for the pre-investment and post-investment problems, respectively. The verification arguments for the pre-investment and post-investment problems are also provided. Finally, numerical simulations are given to illustrate the properties of our model.Optimal chemotherapy for brain tumor growth in a reaction-diffusion modelhttps://zbmath.org/1472.920982021-11-25T18:46:10.358925Z"Yousefnezhad, Mohsen"https://zbmath.org/authors/?q=ai:yousefnezhad.mohsen"Kao, Chiu-Yen"https://zbmath.org/authors/?q=ai:kao.chiu-yen"Mohammadi, Seyyed Abbas"https://zbmath.org/authors/?q=ai:mohammadi.seyyed-abbasOptimizing the usability of brain-computer interfaceshttps://zbmath.org/1472.921302021-11-25T18:46:10.358925Z"Zhang, Yin"https://zbmath.org/authors/?q=ai:zhang.yin|zhang.yin.1"Chase, Steve M."https://zbmath.org/authors/?q=ai:chase.steve-mSummary: Brain-computer interfaces are in the process of moving from the laboratory to the clinic. These devices act by reading neural activity and using it to directly control a device, such as a cursor on a computer screen. An open question in the field is how to map neural activity to device movement in order to achieve the most proficient control. This question is complicated by the fact that learning, especially the long-term skill learning that accompanies weeks of practice, can allow subjects to improve performance over time. Typical approaches to this problem attempt to maximize the biomimetic properties of the device in order to limit the need for extensive training. However, it is unclear if this approach would ultimately be superior to performance that might be achieved with a nonbiomimetic device once the subject has engaged in extended practice and learned how to use it. Here we approach this problem using ideas from optimal control theory. Under the assumption that the brain acts as an optimal controller, we present a formal definition of the usability of a device and show that the optimal postlearning mapping can be written as the solution of a constrained optimization problem. We then derive the optimal mappings for particular cases common to most brain-computer interfaces. Our results suggest that the common approach of creating biomimetic interfaces may not be optimal when learning is taken into account. More broadly, our method provides a blueprint for optimal device design in general control-theoretic contexts.Vaccination in a two-group epidemic modelhttps://zbmath.org/1472.921352021-11-25T18:46:10.358925Z"Aniţa, Sebastian"https://zbmath.org/authors/?q=ai:anita.sebastian"Banerjee, Malay"https://zbmath.org/authors/?q=ai:banerjee.malay"Ghosh, Samiran"https://zbmath.org/authors/?q=ai:ghosh.samiran|ghosh.samiran.1"Volpert, Vitaly"https://zbmath.org/authors/?q=ai:volpert.vitaly-aSummary: Epidemic progression depends on the structure of the population. We study a two-group epidemic model with the difference between the groups determined by the rate of disease transmission. The basic reproduction number, the maximal and the total number of infected individuals are characterized by the proportion between the groups. We consider different vaccination strategies and determine the outcome of the vaccination campaign depending on the distribution of vaccinated individuals between the groups.How to coordinate vaccination and social distancing to mitigate SARS-CoV-2 outbreakshttps://zbmath.org/1472.921372021-11-25T18:46:10.358925Z"Grundel, Sara M."https://zbmath.org/authors/?q=ai:grundel.sara-m"Heyder, Stefan"https://zbmath.org/authors/?q=ai:heyder.stefan"Hotz, Thomas"https://zbmath.org/authors/?q=ai:hotz.thomas"Ritschel, Tobias K. S."https://zbmath.org/authors/?q=ai:ritschel.tobias-k-s"Sauerteig, Philipp"https://zbmath.org/authors/?q=ai:sauerteig.philipp"Worthmann, Karl"https://zbmath.org/authors/?q=ai:worthmann.karlAn optimal control study with quantity of additional food as control in prey-predator systems involving inhibitory effecthttps://zbmath.org/1472.921702021-11-25T18:46:10.358925Z"Ananth, V. S."https://zbmath.org/authors/?q=ai:ananth.v-s"Vamsi, D. K. K."https://zbmath.org/authors/?q=ai:vamsi.dasu-krishna-kiranSummary: Additional food provided prey-predator systems have become a significant and important area of study for both theoretical and experimental ecologists. This is mainly because provision of additional food to the predator in the prey-predator systems has proven to facilitate wildlife conservation as well as reduction of pesticides in agriculture. Further, the mathematical modeling and analysis of these systems provide the eco-manager with various strategies that can be implemented on field to achieve the desired objectives. The outcomes of many theoretical and mathematical studies of such additional food systems have shown that the quality and quantity of additional food play a crucial role in driving the system to the desired state. However, one of the limitations of these studies is that they are asymptotic in nature, where the desired state is reached eventually with time. To overcome these limitations, we present a time optimal control study for an additional food provided prey-predator system involving inhibitory effect with quantity of additional food as the control parameter with the objective of reaching the desired state in finite (minimum) time. The results show that the optimal solution is a bang-bang control with a possibility of multiple switches. Numerical examples illustrate the theoretical findings. These results can be applied to both biological conservation and pest eradication.Optimal virulence, diffusion and tradeoffshttps://zbmath.org/1472.922052021-11-25T18:46:10.358925Z"da Silva, Esdras Jafet Aristides"https://zbmath.org/authors/?q=ai:da-silva.esdras-jafet-aristides"Castilho, César"https://zbmath.org/authors/?q=ai:castilho.cesarIn this paper, the authors put forward the idea of an alternative of a classical SIR epidemiological model in which pathogens are identified by a (phenotypic) mutant trait \(x\). And assume that the trait \(x\) directly influences the epidemiological components of the pathogen, particularly the infection rate function \(\beta(x)\), the clearance rate \(\gamma(x)\) and the disease induced mortality \(\alpha(x)\). The main objective of this study is to find the phenotypic functions, that maximize the pathogen fitness. By analysing the principle of maximizing the basic reproductive number of the pathogen, authors discussed its evolutionary development as an optimal control problem -- pathogen's possible optimal evolutionary strategies were recognized with the help of Pontryagin's maximum principle. In the circumstances of virulence evolution, 3 types of optimal evolutionary routes were identified and analysed qualitatively. Each optimal solution forces a different tradeoff relation among the epidemiological parameters. In this study, the authors forecast two kinds of infections: short-lasting mild infections \& long-lasting acute infections.The complex dynamics of hepatitis B infected individuals with optimal controlhttps://zbmath.org/1472.922082021-11-25T18:46:10.358925Z"Din, Anwarud"https://zbmath.org/authors/?q=ai:din.anwarud"Li, Yongjin"https://zbmath.org/authors/?q=ai:li.yongjin"Shah, Murad Ali"https://zbmath.org/authors/?q=ai:shah.murad-aliSummary: This paper proposes various stages of the hepatitis B virus (HBV) besides its transmissibility and nonlinear incidence rate to develop an epidemic model. The authors plan the model, and then prove some basic results for the well-posedness in term of boundedness and positivity. Moreover, the authors find the threshold parameter \(R_0\), called the basic/effective reproductive number and carry out local sensitive analysis. Furthermore, the authors examine stability and hence condition for stability in terms of \(R_0\). By using sensitivity analysis, the authors formulate a control problem in order to eradicate HBV from the population and proved that the control problem actually exists. The complete characterization of the optimum system was achieved by using the \(4^{\mathrm{th}}\)-order Runge-Kutta procedure.Stochastic chlamydia dynamics and optimal spreadhttps://zbmath.org/1472.922102021-11-25T18:46:10.358925Z"Enciso, German"https://zbmath.org/authors/?q=ai:enciso.german-andres"Sütterlin, Christine"https://zbmath.org/authors/?q=ai:sutterlin.christine"Tan, Ming"https://zbmath.org/authors/?q=ai:tan.ming"Wan, Frederic Y. M."https://zbmath.org/authors/?q=ai:wan.frederic-yui-mingSummary: \textit{Chlamydia trachomatis} is an important bacterial pathogen that has an unusual developmental switch from a dividing form (reticulate body or RB) to an infectious form (elementary body or EB). RBs replicate by binary fission within an infected host cell, but there is a delay before RBs convert into EBs for spread to a new host cell. We developed stochastic optimal control models of the Chlamydia developmental cycle to examine factors that control the number of EBs produced. These factors included the probability and timing of conversion, and the duration of the developmental cycle before the host cell lyses. Our mathematical analysis shows that the observed delay in RB-to-EB conversion is important for maximizing EB production by the end of the intracellular infection.Optimal control for a COVID-19 model accounting for symptomatic and asymptomatichttps://zbmath.org/1472.922262021-11-25T18:46:10.358925Z"Macalisang, Jead M."https://zbmath.org/authors/?q=ai:macalisang.jead-m"Caay, Mark L."https://zbmath.org/authors/?q=ai:caay.mark-l"Arcede, Jayrold P."https://zbmath.org/authors/?q=ai:arcede.jayrold-p"Caga-anan, Randy L."https://zbmath.org/authors/?q=ai:caga-anan.randy-lSummary: Building on an SEIR-type model of COVID-19 where the infected persons are further divided into symptomatic and asymptomatic, a system incorporating the various possible interventions is formulated. Interventions, also referred to as controls, include transmission reduction (e.g., lockdown, social distancing, barrier gestures); testing/isolation on the exposed, symptomatic and asymptomatic compartments; and medical controls such as enhancing patients' medical care and increasing bed capacity. By considering the government's capacity, the best strategies for implementing the controls were obtained using optimal control theory. Results show that, if all the controls are to be used, the more able the government is, the more it should implement transmission reduction, testing, and enhancing patients' medical care without increasing hospital beds. However, if the government finds it very difficult to implement the controls for economic reasons, the best approach is to increase the hospital beds. Moreover, among the testing/isolation controls, testing/isolation in the exposed compartment is the least needed when there is significant transmission reduction control. Surprisingly, when there is no transmission reduction control, testing/isolation in the exposed should be optimal. Testing/isolation in the exposed could seemingly replace the transmission reduction control to yield a comparable result to that when the transmission reduction control is being implemented.A co-infection model for oncogenic human papillomavirus and tuberculosis with optimal control and cost-effectiveness analysishttps://zbmath.org/1472.922342021-11-25T18:46:10.358925Z"Omame, Andrew"https://zbmath.org/authors/?q=ai:omame.andrew"Okuonghae, Daniel"https://zbmath.org/authors/?q=ai:okuonghae.danielSummary: A co-infection model for oncogenic human papillomavirus (HPV) and tuberculosis (TB), with optimal control and cost-effectiveness analysis is studied and analyzed to assess the impact of controls against incident infection and against infection with HPV by TB-infected individuals as well as optimal TB treatment in reducing the burden of the co-infection of the two diseases in a population. The co-infection model exhibits backward bifurcation when the associated reproduction number is less than unity. Furthermore, it is shown that TB and HPV re-infection parameters \((\varphi_p \neq 0\) and \(\sigma_t \neq 0)\) as well as TB exogenous re-infection term \((\epsilon_1 \neq 0)\) induced the phenomenon of backward bifurcation in the oncogenic HPV-TB co-infection model. The global asymptotic stability of the disease-free equilibrium of the co-infection model is \textit{shown not to exist}, when the associated reproduction number is below unity. The necessary conditions for the existence of optimal control and the optimality system for the co-infection model is established using the Pontryagin's maximum principle. Numerical simulations of the optimal control model reveal that the intervention strategy which combines and implements control against HPV infection by TB infected individuals as well as TB treatment control for dually infected individuals is the most cost-effective of all the control strategies for the control and management of the burden of oncogenic HPV and TB co-infection.Optimal control of a Nipah virus transmission modelhttps://zbmath.org/1472.922362021-11-25T18:46:10.358925Z"Panja, Prabir"https://zbmath.org/authors/?q=ai:panja.prabir"Jana, Ranjan Kumar"https://zbmath.org/authors/?q=ai:jana.ranjan-kumarSummary: In this chapter, a Nipah virus transmission model in human population has been formulated. According to the transmission mechanism of Nipah virus, it is assumed that the Nipah virus is transmitted in human population in two possible ways: (i) through infected pig and (ii) through infected human. Optimal control strategies have been developed to minimize the number of infected humans from Nipah virus as well as to minimize the cost of control strategies. Boundedness of solutions and different possible equilibrium points have been studied. The existence condition of the optimal control problem has been investigated. The numerical simulation results without control and with control (vaccination and treatment) of Nipah virus transmission model have been compared. From the theoretical and numerical results, it can be concluded that in order to reduce Nipah virus transmission from human population, vaccination and treatment controls can be considered together.
For the entire collection see [Zbl 1461.92002].Controllability and optimal control for a class of time-delayed fractional stochastic integro-differential systemshttps://zbmath.org/1472.930162021-11-25T18:46:10.358925Z"Sathiyaraj, T."https://zbmath.org/authors/?q=ai:sathiyaraj.t"Wang, JinRong"https://zbmath.org/authors/?q=ai:wang.jinrong"Balasubramaniam, P."https://zbmath.org/authors/?q=ai:balasubramaniam.pagavathigounderSummary: In this paper, we study the controllability and optimal control for a class of time-delayed fractional stochastic integro-differential system with Poisson jumps. A set of sufficient conditions is established for complete and approximate controllability by assuming non-Lipschitz conditions and \textit{pth} mean square norm. We also give an existence of optimal control for Bolza problem. Our result is valid for fractional order \(\alpha >\frac{p-1}{p}\), \(p\ge 2.\) Finally, an example is provided to illustrate the efficiency of the obtained theoretical results.Multiple model unfalsified adaptive generalized predictive control based on the quadratic inverse optimal control concepthttps://zbmath.org/1472.930372021-11-25T18:46:10.358925Z"Forouz, Bahman Sadeghi"https://zbmath.org/authors/?q=ai:forouz.bahman-sadeghi"Manzar, Mojtaba Nouri"https://zbmath.org/authors/?q=ai:nouri-manzar.mojtaba"Khaki-Sedigh, Ali"https://zbmath.org/authors/?q=ai:khaki-sedigh.aliSummary: Unfalsified adaptive control (UAC) is a class of switching control systems which deals with the control of uncertain systems. The UAC includes a bank of controllers, a supervisor, and a system in which the supervisor selects a stabilizing controller based on the system input and output data. Feasibility is the only assumption required in the UAC strategy, which guarantees that there is at least one stabilizing controller in the controller bank. UAC uses the cost detectability definition to prove closed-loop stability. The combination of UAC and multiple model supervisory adaptive control (MMASC) results in the proposed unfalsified multi-model control methodology that enjoys appropriate transient performance and stability proof with required minimum assumptions. In practical controller implementations, the effect of actuator constraints on the control signals is crucial. Despite the significance of constrained systems analysis in real applications, the input constraints in the structures of unfalsified control are not generally considered. Also, the stability analysis of constrained unfalsified control is key to its practical applications. In this article, the input constrained systems are considered using the constrained generalized predictive control (GPC) as the main controllers. Subsequently, to handle virtual signal generation for the GPCs in the UAC context, the inverse optimal control strategy is engaged and formulated to solve the signal generation problem. Simulation results are employed to show the effectiveness of the proposed methodology.Synthesis of control Lyapunov functions and stabilizing feedback strategies using exit-time optimal control. I: Theoryhttps://zbmath.org/1472.930752021-11-25T18:46:10.358925Z"Yegorov, Ivan"https://zbmath.org/authors/?q=ai:yegorov.ivan-v"Dower, Peter M."https://zbmath.org/authors/?q=ai:dower.peter-m"Grüne, Lars"https://zbmath.org/authors/?q=ai:grune.larsSummary: This work studies the problem of constructing control Lyapunov functions (CLFs) and feedback stabilization strategies for deterministic nonlinear control systems described by ordinary differential equations. Many numerical methods for solving the Hamilton-Jacobi-Bellman partial differential equations specifying CLFs typically require dense state space discretizations and consequently suffer from the curse of dimensionality. A relevant direction of attenuating the curse of dimensionality concerns reducing the computation of the values of CLFs and associated feedbacks at any selected states to finite-dimensional nonlinear programming problems. We propose to use exit-time optimal control for that purpose. This article is the first part of a two-part work. First, we state an exit-time optimal control problem with respect to a sublevel set of an appropriate local CLF and establish that, under a number of reasonable conditions, the concatenation of the corresponding value function and the local CLF is a global CLF in the whole domain of asymptotic null-controllability. We also investigate the formulated optimal control problem. A modification of these constructions for the case when one does not find a suitable local CLF is provided as well. Our developments serve as a theoretical basis for a curse-of-dimensionality-free approach to feedback stabilization, that is presented in the second part of our work [ibid. 42, No. 5, 1410--1440 (2021; Zbl 1472.93076)] together with supporting numerical simulation results.Synthesis of control Lyapunov functions and stabilizing feedback strategies using exit-time optimal control. II: Numerical approachhttps://zbmath.org/1472.930762021-11-25T18:46:10.358925Z"Yegorov, Ivan"https://zbmath.org/authors/?q=ai:yegorov.ivan-v"Dower, Peter M."https://zbmath.org/authors/?q=ai:dower.peter-m"Grüne, Lars"https://zbmath.org/authors/?q=ai:grune.larsSummary: This paper continues our study [the authors, ibid. 42, No. 5, 1385--1409 (2021; Zbl 1472.93075)] and develops a curse-of-dimensionality-free numerical approach to feedback stabilization, whose theoretical foundation was built in [loc. cit.] and involved the characterization of control Lyapunov functions (CLFs) via exit-time optimal control. First, we describe an auxiliary linearization-based technique for the construction of a local CLF and discuss how to determine its appropriate sublevel set that can serve as the terminal set in the exit-time optimal control problem leading to a global or semi-global CLF. Next, the curse of complexity is addressed with regard to the approximation of CLFs and associated feedback strategies in high-dimensional regions. The goal is to enable for efficient model predictive control implementations with essentially faster (though less accurate) online policy updates than in case of solving direct or characteristics-based nonlinear programming problems for each sample instant. We propose a computational approach that combines gradient enhanced modifications of the Kriging and inverse distance weighting frameworks for scattered grid interpolation. It in particular allows for convenient offline inclusion of new data to improve obtained approximations (machine learning can be used to select relevant new sparse grid nodes). Moreover, our method is designed so as to a priori return proper values of the CLF interpolant and its gradient on the entire terminal set of the considered exit-time optimal control problem. Supporting numerical simulation results are also presented.An inverse optimal approach to ship course-keeping controlhttps://zbmath.org/1472.930972021-11-25T18:46:10.358925Z"Wang, Chuanrui"https://zbmath.org/authors/?q=ai:wang.chuanrui"Yan, Chuanxu"https://zbmath.org/authors/?q=ai:yan.chuanxu"Liu, Zhenchong"https://zbmath.org/authors/?q=ai:liu.zhenchong"Cao, Feng"https://zbmath.org/authors/?q=ai:cao.fengSummary: This paper deals with the ship course tracking control problem in a novel inverse optimal control approach. The inverse optimal stabilization problem and inverse optimal gain assignment problem are firstly extended to general systems affine in the control with unknown control gain. It is shown that a sufficient condition to solve the inverse optimal control problem is the existence of a stabilization control law in a special form for a corresponding auxiliary system. Then, by employing backstepping technique, control laws are designed which solve the inverse optimal stabilization, inverse optimal adaptive stabilization and inverse optimal adaptive gain assignment problem of ship course control system, respectively. Simulations are included to illustrate the effectiveness of the proposed control algorithms.Event-triggered decentralized optimal fault tolerant control for mismatched interconnected nonlinear systems through adaptive dynamic programminghttps://zbmath.org/1472.931182021-11-25T18:46:10.358925Z"Luo, Fangchao"https://zbmath.org/authors/?q=ai:luo.fangchao"Zhao, Bo"https://zbmath.org/authors/?q=ai:zhao.bo"Liu, Derong"https://zbmath.org/authors/?q=ai:liu.derongSummary: In this paper, we propose an event-triggered decentralized optimal fault tolerant control (ETDOFTC) scheme based on adaptive dynamic programming for mismatched interconnected nonlinear systems with actuator failures. For fault-free dynamic models, the decentralized control problem is addressed by developing a set of decentralized optimal control strategies for isolated subsystems with modified value functions, which are approximated by critic neural networks. Meanwhile, the neural network-based decentralized observer is established to approximate actuator failures and mismatched unknown interconnections. The weights of the neural networks are aperiodically updated at the designed triggering instants. Then, the proposed ETDOFTC scheme is obtained by combining the event-triggered decentralized optimal control strategies with the adaptive fault compensators. Furthermore, it is proved that all the signals of the closed-loop system are uniformly ultimately bounded via the Lyapunov stability analysis. Finally, simulation results of two examples are presented to confirm the effectiveness of the proposed ETDOFTC scheme.A stable reentry trajectory for flexible manipulatorshttps://zbmath.org/1472.931242021-11-25T18:46:10.358925Z"Bastos Jr., Guaraci"https://zbmath.org/authors/?q=ai:bastos.guaraci-junSummary: Trajectory tracking of fast-lightweight manipulators is an emerging topic. These systems often possess an unstable internal dynamics that is related to the inverse dynamics problem. The main contributions of this work are (i) a direct relation between internal dynamics variables and trajectory constraints as state equations in an observer canonical form; (ii) the entry/reentry trajectory defined as input control in order to stabilise the unstable internal dynamics; (iii) the methodology is well-suited for systems modelled in a finite element approach, related to an index-3 differential algebraic equation; (iv) a linearised design is established, but the results are suitable to consider a nonlinear model predictive control of underactuated/flexible manipulators, since a reentry trajectory is often required to minimise the output error recursively. For the application, (a) a planar underactuated manipulator with one passive joint and (b) a planar flexible manipulator on a cart were considered.Modified three-dimensional true proportional navigation and its inverse optimal formhttps://zbmath.org/1472.931582021-11-25T18:46:10.358925Z"Liao, Fei"https://zbmath.org/authors/?q=ai:liao.fei"Zhang, Sheng"https://zbmath.org/authors/?q=ai:zhang.shengSummary: A novel three-dimensional modified true proportional navigation (MTPN) guidance and its inverse optimal form are proposed for the interception of the nonmaneuvering and maneuvering targets, and the nonlinear dynamic characteristics of pursuit situations and target maneuvers are taken into full account. The proposed approach is feasible without knowing any prior information or estimate of the target maneuvers, which are regarded as the unknown disturbance input. Therefore, the TPN-type guidance is extended to intercept the maneuvering targets with bounded unknown accelerations. Moreover, the explicit physical significance is given for each component of the navigation gain expression of MTPN. The inverse optimal control approach and the input-to-state stability are introduced to the MTPN design. Using the MTPN, the homing missile can conduct the terminal homing guidance phase in any particular direction with respect to their nonmaneuvering or maneuvering targets through choosing an appropriate inertial reference coordinate system in practice. Using the proposed schemes, the line-of-sight (LOS) rate is globally exponentially stable for nonmaneuvering targets, and is input-to-state stability (ISS) for maneuvering targets with bounded unknown accelerations. Performed simulation results have confirmed the effectiveness of the proposed scheme.Adaptive robust control in continuous timehttps://zbmath.org/1472.931952021-11-25T18:46:10.358925Z"Bhudisaksang, Theerawat"https://zbmath.org/authors/?q=ai:bhudisaksang.theerawat"Cartea, Álvaro"https://zbmath.org/authors/?q=ai:cartea.alvaroThe existence and uniqueness of viscosity solution to a kind of Hamilton-Jacobi-Bellman equationhttps://zbmath.org/1472.931962021-11-25T18:46:10.358925Z"Hu, Mingshang"https://zbmath.org/authors/?q=ai:hu.mingshang"Ji, Shaolin"https://zbmath.org/authors/?q=ai:ji.shaolin"Xue, Xiaole"https://zbmath.org/authors/?q=ai:xue.xiaoleThe authors study the existence and uniqueness of the viscosity solution to a Hamilton-Jacobi-Bellman (HJB) equation coupled with algebra equations. This kind of equation comes from a stochastic optimal control problem for which the control system is governed by a fully coupled forward-backward stochastic differential equation (FBSDE). By extending Peng's backward semigroup, the authors obtain the dynamic programming principle and prove that the value function is a viscosity solution to the HJB equation mentioned above. By the uniqueness of the solution to FBSDEs, they give a probabilistic approach to study the uniqueness of the solution to this HJB equation. They also show that the value function is the minimum viscosity solution to this HJB equation. When the coefficients are independent of the control variable or the solution is smooth, they prove that the value function is the unique viscosity solution.A modified MSA for stochastic control problemshttps://zbmath.org/1472.931982021-11-25T18:46:10.358925Z"Kerimkulov, B."https://zbmath.org/authors/?q=ai:kerimkulov.bekzhan"Šiška, D."https://zbmath.org/authors/?q=ai:siska.david"Szpruch, L."https://zbmath.org/authors/?q=ai:szpruch.lukaszSummary: The classical method of successive approximations (MSA) is an iterative method for solving stochastic control problems and is derived from Pontryagin's optimality principle. It is known that the MSA may fail to converge. Using careful estimates for the backward stochastic differential equation (BSDE) this paper suggests a modification to the MSA algorithm. This modified MSA is shown to converge for general stochastic control problems with control in both the drift and diffusion coefficients. Under some additional assumptions the rate of convergence is shown. The results are valid without restrictions on the time horizon of the control problem, in contrast to iterative methods based on the theory of forward-backward stochastic differential equations.Optimal linear-quadratic-Gaussian control for discrete-time linear systems with white and time-correlated measurement noiseshttps://zbmath.org/1472.932002021-11-25T18:46:10.358925Z"Liu, Wei"https://zbmath.org/authors/?q=ai:liu.wei|liu.wei.3|liu.wei.8|liu.wei.5|liu.wei.6|liu.wei.7|liu.wei.2|liu.wei.9|liu.wei.1"Shi, Peng"https://zbmath.org/authors/?q=ai:shi.peng|shi.peng.1"Xie, Xiangpeng"https://zbmath.org/authors/?q=ai:xie.xiangpeng"Yue, Dong"https://zbmath.org/authors/?q=ai:yue.dong"Fei, Shumin"https://zbmath.org/authors/?q=ai:fei.shuminSummary: This article is concerned with the optimal linear-quadratic-Gaussian control problem for discrete-time linear systems corrupted by white and time-correlated measurement noises. First, an optimal predictor for the system under consideration is proposed. Then, an optimal controller is designed, which minimizes an expected loss by a control strategy where the control is a function of measurement sequence. The novelty of this article is that a new sequence instead of the original measurement sequence is used to obtain the optimal control scheme where the element in the new sequence is derived from measurement differencing. A verification example is given to illustrate the effectiveness of the developed new design method.Infinite horizon multiobjective optimal control of stochastic cooperative linear-quadratic dynamic difference gameshttps://zbmath.org/1472.932012021-11-25T18:46:10.358925Z"Peng, Chenchen"https://zbmath.org/authors/?q=ai:peng.chenchen"Zhang, Weihai"https://zbmath.org/authors/?q=ai:zhang.weihai"Ma, Limin"https://zbmath.org/authors/?q=ai:ma.liminSummary: This article is concerned with the infinite horizon stochastic cooperative linear-quadratic (LQ) dynamic difference game in both the regular and the indefinite cases. Firstly, due to the constraints imposed on the weighting matrices and the linearity of the dynamic system, the costs are shown to be convex spontaneously for the regular stochastic cooperative LQ difference game, which yields the equivalence between the minimization of the weighted sum of costs and the Pareto optimal control. Secondly, the Pareto optimal control is derived for the regular game on the ground of the solution to the weighted algebraic Riccati equation (WARE) under exact observability, and then Pareto solutions are identified via the optimal feedback gain matrices and the solution to the weighted algebraic Lyapunov equation (WALE). Moreover, a new criterion which is also necessary and sufficient is developed to guarantee the costs to be convex for the indefinite case, and the Pareto optimality is investigated based on the solutions to the weighted generalized algebraic Riccati equation (WGARE) and the weighted generalized algebraic Lyapunov equation (WGALE) combining with the semidefinite programming (SDP). Finally, the fishery management game in the economy is presented to illustrate the obtained results.Individual and mass behavior in large population forward-backward stochastic control problems: centralized and Nash equilibrium solutionshttps://zbmath.org/1472.932022021-11-25T18:46:10.358925Z"Wang, Shujun"https://zbmath.org/authors/?q=ai:wang.shujun"Xiao, Hua"https://zbmath.org/authors/?q=ai:xiao.huaSummary: This article studies a class of dynamic optimization of large-population system in which a large number of negligible agents are coupled via state-average in their cost functional and state dynamics. The most significant feature in our setup is the dynamics of individual agents are modeled by forward-backward stochastic differential equations. The associated mean-field game, in its forward-backward sense, is also formulated to seek the decentralized strategies. Unlike the forward case, the consistency conditions of our forward-backward mean-field games involve five Riccati and force rate equations. Moreover, their initial and terminal conditions are mixed thus some special decoupling technique is applied here. We also verify the \(\epsilon\)-Nash equilibrium property of the derived decentralized strategies. To this end, some estimates to backward stochastic system are employed.Non-zero sum differential game for stochastic Markovian jump systems with partially unknown transition probabilitieshttps://zbmath.org/1472.932032021-11-25T18:46:10.358925Z"Zhang, Chengke"https://zbmath.org/authors/?q=ai:zhang.chengke"Li, Fangchao"https://zbmath.org/authors/?q=ai:li.fangchaoSummary: This paper focuses on the non-zero sum differential game problem for Markovian jump systems with partially unknown transition probabilities. Firstly, a suboptimal control problem is studied by the free-connection weighting matrix method, and then the non-zero sum differential game problem is investigated on this basis. Several sufficient conditions for the existence of \(\varepsilon\)-suboptimal control strategy and \(\varepsilon\)-suboptimal Nash equilibrium strategies are provided, and their explicit expressions are designed. Moreover, the precise form for the upper bound of the cost function is also given. To facilitate the calculation, all conditions are converted into the corresponding equivalent linear matrix inequalities or bilinear matrix inequalities form. Finally, two numerical examples are utilized to demonstrate the effectiveness of the main results.A BSDE approach to stochastic linear quadratic control problemhttps://zbmath.org/1472.932042021-11-25T18:46:10.358925Z"Zhang, Wei"https://zbmath.org/authors/?q=ai:zhang.wei.7|zhang.wei.5|zhang.wei.12|zhang.wei.3|zhang.wei.16|zhang.wei.18|zhang.wei.17|zhang.wei.15|zhang.wei.13|zhang.wei.9|zhang.wei.2|zhang.wei.4|zhang.wei.10|zhang.wei.19|zhang.wei.6|zhang.wei.1"Zhang, Liangquan"https://zbmath.org/authors/?q=ai:zhang.liangquanSummary: In this article, we study a kind of linear quadratic optimal control problem driven by forward-backward stochastic differential equations (FBSDEs in short) with deterministic coefficients. The cost functional is defined by the solution of FBSDEs. By means of the Girsanov transformation, the original issue is turned equivalently into the classical LQ problem. By functional analysis approach, some necessary and sufficient conditions for the existence of optimal controls have been obtained. Moreover, we investigate the relationship between two groups of first-order and second-order adjoint equations. A new stochastic Riccati equation is derived, which leads to the state feedback form of optimal control. By introducing a new Hamiltonian function with an exponential factor, we establish the stochastic maximum principle to deal with the stochastic linear quadratic problem for forward-backward stochastic system with nonconvex control domain using first-order adjoint equation. An illustrative example is given as well.Phase retrieval from Fourier measurements with maskshttps://zbmath.org/1472.940092021-11-25T18:46:10.358925Z"Li, Huiping"https://zbmath.org/authors/?q=ai:li.huiping"Li, Song"https://zbmath.org/authors/?q=ai:li.songSummary: This paper concerns the problem of phase retrieval from Fourier measurements with random masks. Here we focus on researching two kinds of random masks. Firstly, we utilize the Fourier measurements with real masks to estimate a general signal \(\mathfrak{x}_0\in \mathbb{R}^d\) in noiseless case when \(d\) is even. It is demonstrated that \(O(\log^2d)\) real random masks are able to ensure accurate recovery of \(\mathfrak{x}_0\). Then we find that such real masks are not adaptable to reconstruct complex signals of even dimension. Subsequently, we prove that \(O(\log^4d)\) complex masks are enough to stably estimate a general signal \(\mathfrak{x}_0\in \mathbb{C}^d\) under bounded noise interference, which extends \textit{E. J. Candès} et al.'s work [SIAM Rev. 57, No. 2, 225--251 (2015; Zbl 1344.49057)]. Meanwhile, we establish tighter error estimations for real signals of even dimensions or complex signals of odd dimensions by using \(O(\log^2d)\) real masks. Finally, we intend to tackle with the noisy phase problem about an \(s\)-sparse signal by a robust and efficient approach, namely, two-stage algorithm. Based on the stable guarantees for general signals, we show that the \(s\)-sparse signal \(\mathfrak{x}_0\) can be stably recovered from composite measurements under near-optimal sample complexity up to a \(\log\) factor, namely, \(O(s\log(\frac{ed}{s})\log^4(s\log(\frac{ed}{s})))\).