Recent zbMATH articles in MSC 49Jhttps://zbmath.org/atom/cc/49J2022-09-13T20:28:31.338867ZUnknown authorWerkzeugBook review of: L. Ambrosio et al., Lectures on optimal transporthttps://zbmath.org/1491.000312022-09-13T20:28:31.338867Z"Thorpe, Matthew"https://zbmath.org/authors/?q=ai:thorpe.matthewReview of [Zbl 1485.49001].Second order state-dependent sweeping process with unbounded perturbationhttps://zbmath.org/1491.340332022-09-13T20:28:31.338867Z"Affane, Doria"https://zbmath.org/authors/?q=ai:affane.doria"Fetouci, Nora"https://zbmath.org/authors/?q=ai:fetouci.nora"Yarou, Mustapha Fateh"https://zbmath.org/authors/?q=ai:yarou.mustapha-fatehSummary: We establish, in the setting of an infinite dimensional Hilbert space, results concerning the existence of solutions of second order ``nonconvex sweeping process'' for a class of uniformly prox-regular sets depending on time and state. The perturbation considered here is general and takes the form of a sum of a single-valued Carathéodory mapping and a set-valued unbounded mapping. We deal also with a delayed perturbation, that is the external forces applied on the system in presence of a finite delay. We extend a discretization approach known for the time-dependent case to the time and state-dependent sweeping process.Remarks on the vanishing viscosity process of state-constraint Hamilton-Jacobi equationshttps://zbmath.org/1491.350142022-09-13T20:28:31.338867Z"Han, Yuxi"https://zbmath.org/authors/?q=ai:han.yuxi"Tu, Son N. T."https://zbmath.org/authors/?q=ai:tu.son-n-tSummary: We investigate the convergence rate in the vanishing viscosity process of the solutions to the subquadratic state-constraint Hamilton-Jacobi equations. We give two different proofs of the fact that, for non-negative Lipschitz data that vanish on the boundary, the rate of convergence is \(\mathcal{O}(\sqrt{\varepsilon})\) in the interior. Moreover, the one-sided rate can be improved to \(\mathcal{O}(\varepsilon)\) for non-negative compactly supported data and \(\mathcal{O}(\varepsilon^{1/p})\) (where \(1<p<2\) is the exponent of the gradient term) for non-negative data \(f\in \mathrm{C}^2 (\overline{\varOmega})\) such that \(f = 0\) and \(Df = 0\) on the boundary. Our approach relies on deep understanding of the blow-up behavior near the boundary and semiconcavity of the solutions.Optimal control for self-organizing target detection model in the 1d casehttps://zbmath.org/1491.351212022-09-13T20:28:31.338867Z"Ryu, Sang-Uk"https://zbmath.org/authors/?q=ai:ryu.sang-ukSummary: This paper is concerned with the optimal control problem associated to the self-organizing target detection model in 1D domains. That is, we show the global existence of weak solution and the existence of optimal control.Error estimates for a pointwise tracking optimal control problem of a semilinear elliptic equationhttps://zbmath.org/1491.351962022-09-13T20:28:31.338867Z"Allendes, Alejandro"https://zbmath.org/authors/?q=ai:allendes.alejandro"Fuica, Francisco"https://zbmath.org/authors/?q=ai:fuica.francisco"Otárola, Enrique"https://zbmath.org/authors/?q=ai:otarola.enriqueTime optimal control problem of the 3D Navier-Stokes-\( \alpha\) equationshttps://zbmath.org/1491.353212022-09-13T20:28:31.338867Z"Son, Dang Thanh"https://zbmath.org/authors/?q=ai:son.dang-thanh"Thuy, Le Thi"https://zbmath.org/authors/?q=ai:thuy.le-thi-hong|le-thi-thuy.Summary: In this paper, we study an optimal control problem for the three-dimensional Navier-Stokes-\(\alpha\) equations in bounded domains with Dirichlet boundary conditions, where the time needed to reach a desired state plays an essential role. We first prove the existence of optimal solutions. Then we establish the first-order and second-order necessary optimality conditions, and the second-order sufficient optimality conditions. The second-order optimality ones obtained in the paper seem to be optimal in the sense that the gap between them is minimal.Nonrelativistic limit of ground state solutions for nonlinear Dirac-Klein-Gordon systemshttps://zbmath.org/1491.353562022-09-13T20:28:31.338867Z"Dong, Xiaojing"https://zbmath.org/authors/?q=ai:dong.xiaojing"Tang, Zhongwei"https://zbmath.org/authors/?q=ai:tang.zhongweiSummary: We study the nonrelativistic limit and some properties of the solutions
\[
(\psi,\phi):=(u,v,\phi) \in \mathbb{C}^2 \times \mathbb{C}^2 \times \mathbb{R}
\]
for the following nonlinear Dirac-Klein-Gordon systems:
\[
\begin{cases}
ic \displaystyle \sum_{k=1}^{3}a_k \partial_k \psi - mc^2 \beta \psi - \omega\psi -\lambda\phi\beta\psi = |\psi|^{p-2}\psi, \\
-\Delta\phi +c^2M^2\phi = 4\pi\lambda(\beta\psi) \cdot \psi,
\end{cases}
\]
where \(p \in [\frac{12}{5},\frac{8}{3}]\), \(c\) denotes the speed of light, \(m > 0\) is the mass of the electron. We show that the first component \(u\) and the last one \(\phi\) of ground state solutions for nonlinear Dirac-Klein-Gordon systems converge to zero and the second one \(v\) converges to corresponding solutions of a coupled system of nonlinear Schrödinger equations as the speed of light tends to infinity for electrons with small mass. Moreover, we also prove the uniform boundedness and the exponential decay properties
of the solutions for the nonlinear Dirac-Klein-Gordon systems with respect to the speed of light \(c\).Maps that are continuously differentiable in the Michal and Bastiani sense but not in the Fréchet sensehttps://zbmath.org/1491.460292022-09-13T20:28:31.338867Z"Walther, H.-O."https://zbmath.org/authors/?q=ai:walther.hans-ottoSummary: We construct examples of nonlinear maps on function spaces which are continuously differentiable in the sense of Michal and Bastiani but not in the sense of Fréchet. The search for such examples is motivated by studies of delay differential equations with the delay variable and not necessarily bounded.Graph convergence and generalized Yosida approximation operator with an applicationhttps://zbmath.org/1491.470552022-09-13T20:28:31.338867Z"Ahmad, Rais"https://zbmath.org/authors/?q=ai:ahmad.rais"Ishtyak, Mohd."https://zbmath.org/authors/?q=ai:ishtyak.mohd"Rahaman, Mijanur"https://zbmath.org/authors/?q=ai:rahaman.mijanur"Ahmad, Iqbal"https://zbmath.org/authors/?q=ai:ahmad.iqbalSummary: In this paper, we introduce a Yosida inclusion problem as well as a generalized Yosida approximation operator. Using the graph convergence of \(H(\cdot, \cdot)\)-accretive operator and resolvent operator convergence discussed in [\textit{X. Li} and \textit{N.-j. Huang}, Appl. Math. Comput. 217, No. 22, 9053--9061 (2011; Zbl 1308.47072)], we establish the convergence for generalized Yosida approximation operator. As an application, we solve a Yosida inclusion problem in \(q\)-uniformly smooth Banach spaces. An example is constructed, and through \texttt{MATLAB} programming, we show some graphics for the convergence of generalized Yosida approximation operator.Composite implicit viscosity extragradient algorithms for systems of variational inequalities with fixed point constraints of asymptotically nonexpansive mappingshttps://zbmath.org/1491.470572022-09-13T20:28:31.338867Z"Ceng, Lu-Chuan"https://zbmath.org/authors/?q=ai:ceng.lu-chuanSummary: A composite implicit viscosity extragradient method based on Korpelevich's extragradient method, implicit viscosity approximation method, and Mann's iteration method is studied and we consider a general system of variational inequalities and a common fixed point problem of an asymptotically nonexpansive mapping and countably many nonexpansive mappings in real Hilbert spaces.Split systems of general nonconvex variational inequalities and fixed point problemshttps://zbmath.org/1491.470582022-09-13T20:28:31.338867Z"Chen, Jin-Zuo"https://zbmath.org/authors/?q=ai:chen.jinzuoSummary: The purpose of this paper is to introduce and study split systems of general nonconvex variational inequalities. Taking advantage of the projection technique over uniformly prox-regularity sets and utilizing two nonlinear operators, we propose and analyze an iterative scheme for solving the split systems of general nonconvex variational inequalities and fixed point problems. We prove that the sequence generated by the suggested iterative algorithm converges strongly to a common solution of the foregoing split problem and fixed point problem. The result presented in this paper extends and improves some well-known results in the literature. Numerical example illustrates the theoretical result.Extragradient methods for solving equilibrium problems, variational inequalities, and fixed point problemshttps://zbmath.org/1491.470642022-09-13T20:28:31.338867Z"Jouymandi, Zeynab"https://zbmath.org/authors/?q=ai:jouymandi.zeynab"Moradlou, Fridoun"https://zbmath.org/authors/?q=ai:moradlou.fridounSummary: In this paper, we propose the new extragradient algorithms for an \(\alpha\)-inverse-strongly monotone operator and a relatively nonexpansive mapping in Banach spaces. We prove convergence theorems by this methods under suitable conditions. Applying our algorithms, we find a zero point of maximal monotone operators. Using \texttt{FMINCON} optimization toolbox in \texttt{MATLAB}, we give an example to illustrate the usability of our results.Strong convergence of a double projection-type method for monotone variational inequalities in Hilbert spaceshttps://zbmath.org/1491.470652022-09-13T20:28:31.338867Z"Kanzow, Christian"https://zbmath.org/authors/?q=ai:kanzow.christian"Shehu, Yekini"https://zbmath.org/authors/?q=ai:shehu.yekiniSummary: We introduce a projection-type algorithm for solving monotone variational inequality problems in real Hilbert spaces without assuming Lipschitz continuity of the corresponding operator. We prove that the whole sequence of iterates converges strongly to a solution of the variational inequality. The method uses only two projections onto the feasible set in each iteration in contrast to other strongly convergent algorithms which either require plenty of projections within a step size rule or have to compute projections on possibly more complicated sets. Some numerical results illustrate the behavior of our method.Common solution to a split equality monotone variational inclusion problem, a split equality generalized general variational-like inequality problem and a split equality fixed point problemhttps://zbmath.org/1491.470662022-09-13T20:28:31.338867Z"Kazmi, K. R."https://zbmath.org/authors/?q=ai:kazmi.kaleem-raza"Ali, Rehan"https://zbmath.org/authors/?q=ai:ali.rehan"Furkan, Mohd"https://zbmath.org/authors/?q=ai:furkan.mohdSummary: This paper deals with a strong convergence theorem for an iterative method for approximating a common solution to a split equality monotone variational inclusion problem, a split equality generalized general variational-like inequality problem and a split equality fixed point problem for quasi-nonexpansive mappings in real Hilbert spaces. Some consequences are derived from the main result. Finally, we give a numerical example to justify the main result. The main result extends and unifies some recent known results in the literature.Iterative algorithm for split generalized mixed equilibrium problem involving relaxed monotone mappings in real Hilbert spaceshttps://zbmath.org/1491.470682022-09-13T20:28:31.338867Z"Osisiogu, Ugochukwu Anulobi"https://zbmath.org/authors/?q=ai:osisiogu.ugochukwu-anulobi"Adum, Friday Lawrence"https://zbmath.org/authors/?q=ai:adum.friday-lawrence"Izuchukwu, Chinedu"https://zbmath.org/authors/?q=ai:izuchukwu.chineduSummary: The main purpose of this paper is to introduce a certain class of split generalized mixed equilibrium problem involving relaxed monotone mappings. To solve our proposed problem, we introduce an iterative algorithm and obtain its strong convergence to a solution of the split generalized mixed equilibrium problems in Hilbert spaces. As special cases of the proposed problem, we studied the proximal split feasibility problem and variational inclusion problem.An iterative method for equilibrium and constrained convex minimization problemshttps://zbmath.org/1491.470732022-09-13T20:28:31.338867Z"Yazdi, Maryam"https://zbmath.org/authors/?q=ai:yazdi.maryam"Shabani, Mohammad Mehdi"https://zbmath.org/authors/?q=ai:shabani.mohammad-mehdi"Sababe, Saeed Hashemi"https://zbmath.org/authors/?q=ai:hashemi-sababe.saeedSummary: We are concerned with finding a common solution to an equilibrium problem associated with a bifunction, and a constrained convex minimization problem. We propose an iterative fixed point algorithm and prove that the algorithm generates a sequence strongly convergent to a common solution. The common solution is identified as the unique solution of a certain variational inequality.The general split equality problem for Bregman quasi-nonexpansive mappings in Banach spaceshttps://zbmath.org/1491.470742022-09-13T20:28:31.338867Z"Zegeye, Habtu"https://zbmath.org/authors/?q=ai:zegeye.habtuSummary: The purpose of this paper is to propose and study an algorithm for solving the general split equality problem governed by Bregman quasi-nonexpansive mappings in Banach spaces. Under some mild conditions, we established the norm convergence of the proposed algorithm. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.Optimization and control for partial differential equations. Uncertainty quantification, open and closed-loop control, and shape optimizationhttps://zbmath.org/1491.490022022-09-13T20:28:31.338867ZPublisher's description: This book highlights new developments in the wide and growing field of partial differential equations (PDE)-constrained optimization. Optimization problems where the dynamics evolve according to a system of PDEs arise in science, engineering, and economic applications and they can take the form of inverse problems, optimal control problems or optimal design problems. This book covers new theoretical, computational as well as implementation aspects for PDE-constrained optimization problems under uncertainty, in shape optimization, and in feedback control, and it illustrates the new developments on representative problems from a variety of applications.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Bernreuther, Marco; Müller, Georg; Volkwein, Stefan}, Reduced basis model order reduction in optimal control of a nonsmooth semilinear elliptic PDE, 1-32 [Zbl 1490.35153]
\textit{Bottois, Arthur}, Pointwise moving control for the \(1\)-D wave equation, 33-57 [Zbl 1485.93059]
\textit{Gugat, Martin; Herty, Michael}, Limits of stabilizability for a semilinear model for gas pipeline flow, 59-71 [Zbl 1489.76044]
\textit{Almeida, Luis; Arnau, Jesús Bellver; Duprez, Michel; Privat, Yannick}, Minimal cost-time strategies for mosquito population replacement, 73-90 [Zbl 1486.92328]
\textit{Almeida, Luis; Estrada, Jorge; Vauchelet, Nicolas}, The sterile insect technique used as a barrier control against reinfestation, 91-111 [Zbl 1487.35164]
\textit{Herberg, Evelyn; Hinze, Michael}, Variational discretization approach applied to an optimal control problem with bounded measure controls, 113-135 [Zbl 1487.49037]
\textit{Hirn, Adrian; Wollner, Winnifried}, An optimal control problem for equations with p-structure and its finite element discretization, 137-165 [Zbl 1487.49025]
\textit{Langer, Ulrich; Steinbach, Olaf; Tröltzsch, Fredi; Yang, Huidong}, Unstructured space-time finite element methods for optimal sparse control of parabolic equations, 167-188 [Zbl 1487.49004]
\textit{Dirks, Carolin; Wirth, Benedikt}, An adaptive finite element approach for lifted branched transport problems, 189-236 [Zbl 1487.49053]
\textit{Feppon, Florian}, High-order homogenization of the Poisson equation in a perforated periodic domain, 237-281 [Zbl 1487.35042]
\textit{Lemoine, Jérôme; Münch, Arnaud}, Least-squares approaches for the \(2\)D Navier-Stokes system, 285-341 [Zbl 07516518]
\textit{Lance, Gontran; Trélat, Emmanuel; Zuazua, Enrique}, Numerical issues and turnpike phenomenon in optimal shape design, 343-366 [Zbl 1487.49050]
\textit{Marinoschi, Gabriela}, Feedback stabilization of Cahn-Hilliard phase-field systems, 367-392 [Zbl 1485.93452]
\textit{Guth, Philipp A.; Schillings, Claudia; Weissmann, Simon}, Ensemble Kalman filter for neural network-based one-shot inversion, 393-418 [Zbl 1491.65120]
\textit{Opschoor, Joost A. A.; Schwab, Christoph; Zech, Jakob}, Deep learning in high dimension: ReLU neural network expression for Bayesian PDE inversion, 419-462 [Zbl 1487.68200]On generalized Newton's aerodynamic problemhttps://zbmath.org/1491.490032022-09-13T20:28:31.338867Z"Plakhov, A."https://zbmath.org/authors/?q=ai:plakhov.a-yuSummary: We consider the generalized Newton's least resistance problem for convex bodies: minimize the functional \(\iint_\Omega (1 + |\nabla u(x,y)|^2)^{-1} dx\, dy\) in the class of concave functions \(u\colon \Omega \to [0,M]\), where the domain \(\Omega \subset \mathbb{R}^2\) is convex and bounded and \(M > 0\). It has been known (see [\textit{G. Buttazzo} et al., Math. Nachr. 173, 71--89 (1995; Zbl 0835.49001)]) that if \(u\) solves the problem, then \(|\nabla u(x,y)| \ge 1\) at all regular points \((x,y)\) such that \(u(x,y) < M\). We prove that if the upper level set \(L = \{ (x,y)\colon u(x,y) = M \}\) has nonempty interior, then for almost all points of its boundary \((\bar{x}, \bar{y}) \in \partial L\) one has \(\lim_{\substack{(x,y)\to (\bar{x},\bar{y})\\u(x,y)<M}}|\nabla u(x,y)| = 1\). As a by-product, we obtain a result concerning local properties of convex surfaces near ridge points.Optimal distributed control for a coupled phase-field systemhttps://zbmath.org/1491.490042022-09-13T20:28:31.338867Z"Chen, Bosheng"https://zbmath.org/authors/?q=ai:chen.bosheng"Li, Huilai"https://zbmath.org/authors/?q=ai:li.huilai"Liu, Changchun"https://zbmath.org/authors/?q=ai:liu.chein-shanThis paper considered a distributed optimal control for a coupled phase-field system. The control acts in the whole domain and the objective functional is of tracking type. The authors proved the existence of a weak solution to the governing state equantion and the existence of a solution to the nonlinear optimal control problem. The differentiability of the control-to-state mapping was studied which allows to derive a first order necessary optimality condition with the aid of an adjoint state system.
Reviewer: Wei Gong (Beijing)Pontryagin's maximum principle for distributed optimal control of two dimensional tidal dynamics system with state constraints of integral typehttps://zbmath.org/1491.490052022-09-13T20:28:31.338867Z"Mohan, Manil T."https://zbmath.org/authors/?q=ai:mohan.manil-tSummary: In this paper, we consider two dimensional tidal dynamics equations in a bounded domain and address a distributed optimal control problem of minimizing a suitable cost functional with state constraints of integral type (on the velocity field). It is well known that the Pontryagin maximum principle provides the first-order necessary conditions of optimality. We show the existence of an optimal control and establish Pontryagin's maximum principle for the state constrained optimization problem for the tidal dynamics system using Ekeland's variational principle and characterize optimal control through the adjoint variable.Strong solution and optimal control problems for a class of fractional linear equationshttps://zbmath.org/1491.490062022-09-13T20:28:31.338867Z"Plekhanova, M. V."https://zbmath.org/authors/?q=ai:plekhanova.marina-vasilevnas|plekhanova.marina-vasilevnaSummary: In this paper, we examine the unique solvability (in the sense of strong solutions) of the Cauchy problem for a linear inhomogeneous equation in a Banach space solved with respect to the Caputo fractional derivative. We assume that the operator acting on the unknown function in the right-hand side of the equation generates an analytic resolving operator family for the corresponding homogeneous equation. We obtain a representation of a strong solution of the Cauchy problem and examine the solvability of optimal control problems with a convex, lower semicontinuous, lower bounded, coercive functional for the equation considered. The general results obtained are used to prove the existence of an optimal control in problems with specific functionals. Abstract results obtained for a control system described by an equation in a Banach space are illustrated by examples of optimal control problems for a fractional equation whose special cases are the subdiffusion equation and the diffusion wave equation.Quasi-variational problems with non-self map on Banach spaces: existence and applicationshttps://zbmath.org/1491.490072022-09-13T20:28:31.338867Z"Allevi, Elisabetta"https://zbmath.org/authors/?q=ai:allevi.elisabetta"De Giuli, Maria Elena"https://zbmath.org/authors/?q=ai:de-giuli.maria-elena"Milasi, Monica"https://zbmath.org/authors/?q=ai:milasi.monica"Scopelliti, Domenico"https://zbmath.org/authors/?q=ai:scopelliti.domenicoSummary: This paper focuses on the analysis of generalized quasi-variational inequality problems with non-self constraint map. To study such problems, in [\textit{D. Aussel} et al., J. Optim. Theory Appl. 170, No. 3, 818--837 (2016; Zbl 1350.49006)] the authors introduced the concept of the projected solution and proved its existence in finite-dimensional spaces. The main contribution of this paper is to prove the existence of a projected solution for generalized quasi-variational inequality problems with non-self constraint map on real Banach spaces. Then, following the multistage stochastic variational approach introduced in [\textit{R. T. Rockafellar} and \textit{R. J B Wets}, Math. Program. 165, No. 1 (B), 331--360 (2017; Zbl 1378.49010)], we introduce the concept of the projected solution in a multistage stochastic setting, and we prove the existence of such a solution. We apply this theoretical result in studying an electricity market with renewable power sources.Existence results for a class of variational quasi-mixed hemivariational-like inequalitieshttps://zbmath.org/1491.490082022-09-13T20:28:31.338867Z"Mahalik, K."https://zbmath.org/authors/?q=ai:mahalik.k"Nahak, C."https://zbmath.org/authors/?q=ai:nahak.chandalSummary: The paper aims to explore the existence results for a class of variational quasi-mixed hemivariational-like inequality problems with nonlinear terms in reflexive Banach spaces, which contain variational and hemivariational inequalities. We make use of stable \((\eta,\psi)\)-quasimonotonicity, KKM theorem, Clarke's generalized directional derivative and Clarke's generalized gradient to derive the existence theorems for the condition of the constrained set being bounded. Further, we obtain the solution's existence results when the constrained set is unbounded by utilizing suitable coercive conditions. Moreover, we present some sufficient conditions to assure the boundedness of the solutions set. Besides, we also demonstrate a necessary and sufficient criteria for a restricted class of variational quasi-mixed hemivariational-like inequality problems. Several applications of the main results are illustrated. The new developments improve and generalize some well-known works.Well-posedness of constrained evolutionary differential variational-hemivariational inequalities with applicationshttps://zbmath.org/1491.490092022-09-13T20:28:31.338867Z"Migórski, Stanisław"https://zbmath.org/authors/?q=ai:migorski.stanislawSummary: A system of a first order history-dependent evolutionary variational-- hemivariational inequality with unilateral constraints coupled with a nonlinear ordinary differential equation in a Banach space is studied. Based on a fixed point theorem for history dependent operators, results on the well-posedness of the system are proved. Existence, uniqueness, continuous dependence of the solution on the data, and the solution regularity are established. Two applications of dynamic problems from contact mechanics illustrate the abstract results. First application is a unilateral viscoplastic frictionless contact problem which leads to a hemivariational inequality for the velocity field, and the second one deals with a viscoelastic frictional contact problem which is described by a variational inequality.Convergence theorem of relaxed quasimonotone variational inequality problemshttps://zbmath.org/1491.490102022-09-13T20:28:31.338867Z"Kim, Jong Kyu"https://zbmath.org/authors/?q=ai:kim.jong-kyu|kim.jongkyu"Alesemi, Meshari"https://zbmath.org/authors/?q=ai:alesemi.meshari"Salahuddin"https://zbmath.org/authors/?q=ai:salahuddin.anjum-rIn this paper an iterative algorithm is well-developed to solve \(\mu\)-quasimonotone variational inequalities. It is proved that the iterative sequence generated by the algorithm is weakly convergent to the weak solutions of the variational inequalities.
Reviewer: Zijia Peng (Nanning)Nonlinear approximation of 3D smectic liquid crystals: sharp lower bound and compactnesshttps://zbmath.org/1491.490112022-09-13T20:28:31.338867Z"Novack, Michael"https://zbmath.org/authors/?q=ai:novack.michael-r"Yan, Xiaodong"https://zbmath.org/authors/?q=ai:yan.xiaodongSummary: We consider the 3D smectic energy
\[
\mathcal{E}_\varepsilon(u) = \frac{1}{2}\int_\Omega \frac{1}{\varepsilon} \left( \partial_z u-\frac{(\partial_x u)^2+(\partial_y u)^2}{2}\right)^2 +\varepsilon \left(\partial_x^2u + \partial_y^2u\right)^2dx\,dy\,dz.
\]
The model contains as a special case the well-known 2D Aviles-Giga model. We prove a sharp lower bound on \(\mathcal{E}_\varepsilon\) as \(\varepsilon \rightarrow 0\) by introducing 3D analogues of the Jin-Kohn entropies [\textit{W. Jin} and \textit{R. V. Kohn}, J. Nonlinear Sci. 10, No. 3, 355--390 (2000; Zbl 0973.49009)]. The sharp bound corresponds to an equipartition of energy between the bending and compression strains and was previously demonstrated in the physics literature only when the approximate Gaussian curvature of each smectic layer vanishes. Also, for \(\varepsilon_n\rightarrow 0\) and an energy-bounded sequence \(\{u_n\}\) with \(\Vert\nabla u_n\Vert_{L^p(\Omega)}\), \(\Vert \nabla u_n\Vert_{L^2(\partial\Omega)}\le C\) for some \(p>6\), we obtain compactness of \(\nabla u_n\) in \(L^2\) assuming that \(\Delta_{xy}u_n\) has constant sign for each \(n\).Fluctuation estimates for the multi-cell formula in stochastic homogenization of partitionshttps://zbmath.org/1491.490122022-09-13T20:28:31.338867Z"Bach, Annika"https://zbmath.org/authors/?q=ai:bach.annika"Ruf, Matthias"https://zbmath.org/authors/?q=ai:ruf.matthiasSummary: In this paper we derive quantitative estimates in the context of stochastic homogenization for integral functionals defined on finite partitions, where the random surface integrand is assumed to be stationary. Requiring the integrand to satisfy in addition a multiscale functional inequality, we control quantitatively the fluctuations of the asymptotic cell formulas defining the homogenized surface integrand. As a byproduct we obtain a simplified cell formula where we replace cubes by almost flat hyperrectangles.On the finite horizon optimal switching problem with random laghttps://zbmath.org/1491.490132022-09-13T20:28:31.338867Z"Perninge, Magnus"https://zbmath.org/authors/?q=ai:perninge.magnusThis paper solves an optimal switching problem with random lag (modeled by letting the operation mode follow a regime switching Markov-model with transition intensities that depend on the switching mode) with a possibility of component failure (modeled by having absorbing components). The author shows the existence of an optimal control for the problem by applying a probabilistic technique based on the concept of Snell envelopes.
Reviewer: Manuel D. Domínguez de la Iglesia (Ciudad de México)McKean-Vlasov optimal control: the dynamic programming principlehttps://zbmath.org/1491.490182022-09-13T20:28:31.338867Z"Djete, Mao Fabrice"https://zbmath.org/authors/?q=ai:djete.mao-fabrice"Possamaï, Dylan"https://zbmath.org/authors/?q=ai:possamai.dylan"Tan, Xiaolu"https://zbmath.org/authors/?q=ai:tan.xiaoluThe paper is devoted to the problem of optimal control of McKean-Vlasov stochastic differential equations. The authors develope a general theory for the non-Markovian case of McKean-Vlasov stochastic control problem with common noise. They propose and investigate weak and strong formulations of the McKean-Vlasov optimal control problem and show their equivalence. The dynamic programming principle is established for weak and strong formulations as well as for strong formulation where the control is adapted to the common noise. Based on their results for the non-Markovian case, the authors obtain the dynamic programming principle for the Markovian control problem. Associated Hamilton-Jacobi-Bellman equations are derived for the common noise strong formulation and for the general strong formulation.
Reviewer: Svetlana A. Kravchenko (Minsk)Strict convergence with equibounded area and minimal completely vertical liftingshttps://zbmath.org/1491.490342022-09-13T20:28:31.338867Z"Mucci, Domenico"https://zbmath.org/authors/?q=ai:mucci.domenicoSummary: Minimal lifting measures of vector-valued functions of bounded variation were introduced by Jerrard-Jung. They satisfy strong continuity properties with respect to the strict convergence in \(BV\). Moreover, they can be described in terms of the action of the optimal Cartesian currents enclosing the graph of \(u\). We deal with a good notion of completely vertical lifting for maps with values into the two dimensional Euclidean space. We then prove lack of uniqueness in the high codimension case. Relationship with the relaxed area functional in the strict convergence is also discussed.Limits of density-constrained optimal transporthttps://zbmath.org/1491.490362022-09-13T20:28:31.338867Z"Gladbach, Peter"https://zbmath.org/authors/?q=ai:gladbach.peter"Kopfer, Eva"https://zbmath.org/authors/?q=ai:kopfer.evaIn this paper, the authors study two limit cases of constrained optimal transport in \(\mathbb{R}^d\). The problem is understood as a minimisation of a functional on the space of constant-speed curves, where the length of the curve intersected with any given open set is less or equal to the value of an integral of a given function (which may take infinite values) on this open set. The first of the two problems is a model of an infinitesimal membrane, which is derived as a \(\Gamma\)-limit as \(\varepsilon \rightarrow 0\) of optimal transport problems with the constraint of the form \(h^\varepsilon(x) = \alpha \varepsilon\) if \(x_d \in (0,\varepsilon)\) and \(+\infty\) otherwise. The second of the two problems concerns homogenisation of a periodic obstacle, whose size goes down to zero: to be exact, the authors study the \(\Gamma\)-limit as \(\varepsilon \rightarrow 0\) of the optimal transport problem with an obstacle of the form \(h^\varepsilon(x) = h(x/\varepsilon)\), where \(h\) is \(\mathbb{Z}^d\)-periodic. In both cases, the authors describe the \(\Gamma\)-limit in an explicit way and provide some examples.
Reviewer: Wojciech Górny (Warszawa)A self-adaptive Tseng extragradient method for solving monotone variational inequality and fixed point problems in Banach spaceshttps://zbmath.org/1491.650382022-09-13T20:28:31.338867Z"Jolaoso, Lateef Olakunle"https://zbmath.org/authors/?q=ai:jolaoso.lateef-olakunleSummary: In this paper, we introduce a self-adaptive projection method for finding a common element in the solution set of variational inequalities (VIs) and fixed point set for relatively nonexpansive mappings in 2-uniformly convex and uniformly smooth real Banach spaces. We prove a strong convergence result for the sequence generated by our algorithm without imposing a Lipschitz condition on the cost operator of the VIs. We also provide some numerical examples to illustrate the performance of the proposed algorithm by comparing with related methods in the literature. This result extends and improves some recent results in the literature in this direction.Strong convergence of inertial subgradient extragradient algorithm for solving pseudomonotone equilibrium problemshttps://zbmath.org/1491.650402022-09-13T20:28:31.338867Z"Thong, Duong Viet"https://zbmath.org/authors/?q=ai:duong-viet-thong."Cholamjiak, Prasit"https://zbmath.org/authors/?q=ai:cholamjiak.prasit"Rassias, Michael T."https://zbmath.org/authors/?q=ai:rassias.michael-th"Cho, Yeol Je"https://zbmath.org/authors/?q=ai:cho.yeol-jeSummary: In this paper, we propose a new modified subgradient extragradient method for solving equilibrium problems involving pseudomonotone and Lipschitz-type bifunctions in Hilbert spaces. We establish the strong convergence of the proposed method under several suitable conditions. In addition, the linear convergence is obained under strong pseudomonotonicity assumption. Our results generalize and extend some related results in the literature. Finally, we provide numerical experiments to illustrate the performance of the proposed algorithm.Metrically regular mapping and its utilization to convergence analysis of a restricted inexact Newton-type methodhttps://zbmath.org/1491.650502022-09-13T20:28:31.338867Z"Rashid, Mohammed Harunor"https://zbmath.org/authors/?q=ai:rashid.mohammed-harunorSummary: In the present paper, we study the restricted inexact Newton-type method for solving the generalized equation \(0\in f(x)+F(x)\), where \(X\) and \(Y\) are Banach spaces, \(f:X\to Y\) is a Fréchet differentiable function and \(F\colon X\rightrightarrows Y\) is a set-valued mapping with closed graph. We establish the convergence criteria of the restricted inexact Newton-type method, which guarantees the existence of any sequence generated by this method and show this generated sequence is convergent linearly and quadratically according to the particular assumptions on the Fréchet derivative of \(f\). Indeed, we obtain semilocal and local convergence results of restricted inexact Newton-type method for solving the above generalized equation when the Fréchet derivative of \(f\) is continuous and Lipschitz continuous as well as \(f+F\) is metrically regular. An application of this method to variational inequality is given. In addition, a numerical experiment is given which illustrates the theoretical result.Error estimates for a class of discontinuous Galerkin methods for nonsmooth problems via convex duality relationshttps://zbmath.org/1491.651242022-09-13T20:28:31.338867Z"Bartels, Sören"https://zbmath.org/authors/?q=ai:bartels.sorenSummary: We devise and analyze a class of interior penalty discontinuous Galerkin methods for nonlinear and nonsmooth variational problems. Discrete duality relations are derived that lead to optimal error estimates in the case of total-variation regularized minimization or obstacle problems. The analysis provides explicit estimates that precisely determine the role of stabilization parameters. Numerical experiments confirm the optimality of the estimates.A differential variational inequality in the study of contact problems with wearhttps://zbmath.org/1491.740752022-09-13T20:28:31.338867Z"Chen, Tao"https://zbmath.org/authors/?q=ai:chen.tao|chen.tao.1"Huang, Nan-Jing"https://zbmath.org/authors/?q=ai:huang.nan-jing"Sofonea, Mircea"https://zbmath.org/authors/?q=ai:sofonea.mircea|sofonea.mircea-tSummary: We start with a mathematical model which describes the sliding contact of a viscoelastic body with a moving foundation. The contact is frictional and the wear of the contact surfaces is taken into account. We prove that this model leads to a differential variational inequality in which the unknowns are the displacement field and the wear function. Then, inspired by this model, we consider a general differential variational inequality in reflexive Banach spaces, governed by four parameters. We prove the unique solvability of the inequality as well as the continuous dependence of its solution with respect to the parameters. The proofs are based on arguments of monotonicity, compactness, convex analysis and lower semicontinuity. Then, we apply these abstract results to the mathematical model of contact for which we deduce the existence of a unique solution as well as the existence of optimal control for an associate optimal control problem. We also present the corresponding mechanical interpretations.Variational analysis of unilateral contact problem for thermo-piezoelectric materials with frictionhttps://zbmath.org/1491.740762022-09-13T20:28:31.338867Z"Hachlaf, A."https://zbmath.org/authors/?q=ai:hachlaf.abdelhadi"Benaissa, H."https://zbmath.org/authors/?q=ai:benaissa.hicham"Benkhira, EL-H."https://zbmath.org/authors/?q=ai:benkhira.el-hassan"Fakhar, R."https://zbmath.org/authors/?q=ai:fakhar.rachidSummary: This paper deals with the mathematical analysis of quasi-static unilateral contact problem with friction between a thermo-piezoelectric body and a conductive foundation. The material is assumed to have thermo-electro-elastic behavior and the contact is modeled by the Signorini's law, the condition of dry friction and a regularized electrical conductivity condition. The effects of frictional heating and thermal conductivity on the mechanisms of material are taken into account. To prove the existence of a weak solution to the problem, an incremental formulation obtained by using an implicit time scheme is studied. Several estimates on the incremental solutions are given, which allow us to pass to the limit by using compactness results.Gamma-convergence results for nematic elastomer bilayers: relaxation and actuationhttps://zbmath.org/1491.820212022-09-13T20:28:31.338867Z"Cesana, Pierluigi"https://zbmath.org/authors/?q=ai:cesana.pierluigi"León Baldelli, Andrés A."https://zbmath.org/authors/?q=ai:leon-baldelli.andres-aSummary: We compute effective energies of thin bilayer structures composed of soft nematic elastic liquid crystals in various geometrical regimes and functional configurations. Our focus is on elastic foundations composed of an isotropic layer attached to a nematic substrate where order-strain interaction results in complex opto-mechanical instabilities activated \textit{via} coupling through the common interface. Allowing out-of-plane displacements, we compute Gamma-limits for vanishing thickness which exhibit spontaneous stress relaxation and shape-morphing behaviour. This extends the plane strain modelling of the authors [Math. Models Methods Appl. Sci. 28, No. 14, 2863--2904 (2018; Zbl 1411.49008)], and shows the asymptotic emergence of fully coupled active macroscopic nematic foundations. Subsequently, we focus on actuation and compute asymptotic configurations of an active plate on nematic foundation interacting with an applied electric field. From the analytical standpoint, the presence of an electric field and its associated electrostatic work turns the total energy non-convex and non-coercive. We show that equilibrium solutions are min-max points of the system, that min-maximising sequences pass to the limit and, that the limit system can exert mechanical work under applied electric fields.Generic property of the partial calmness condition for bilevel programming problemshttps://zbmath.org/1491.901292022-09-13T20:28:31.338867Z"Ke, Rongzhu"https://zbmath.org/authors/?q=ai:ke.rongzhu"Yao, Wei"https://zbmath.org/authors/?q=ai:yao.wei.1|yao.wei"Ye, Jane J."https://zbmath.org/authors/?q=ai:ye.jane-j"Zhang, Jin"https://zbmath.org/authors/?q=ai:zhang.jin.2Block-coordinate and incremental aggregated proximal gradient methods for nonsmooth nonconvex problemshttps://zbmath.org/1491.901302022-09-13T20:28:31.338867Z"Latafat, Puya"https://zbmath.org/authors/?q=ai:latafat.puya"Themelis, Andreas"https://zbmath.org/authors/?q=ai:themelis.andreas"Patrinos, Panagiotis"https://zbmath.org/authors/?q=ai:patrinos.panagiotisSummary: This paper analyzes block-coordinate proximal gradient methods for minimizing the sum of a separable smooth function and a (nonseparable) nonsmooth function, both of which are allowed to be nonconvex. The main tool in our analysis is the forward-backward envelope, which serves as a particularly suitable continuous and real-valued Lyapunov function. Global and linear convergence results are established when the cost function satisfies the Kurdyka-Łojasiewicz property without imposing convexity requirements on the smooth function. Two prominent special cases of the investigated setting are regularized finite sum minimization and the sharing problem; in particular, an immediate byproduct of our analysis leads to novel convergence results and rates for the popular Finito/MISO algorithm in the nonsmooth and nonconvex setting with very general sampling strategies.Optimality and duality for weak quasi efficiency of multiobjective fractional problems via convexificatorshttps://zbmath.org/1491.901532022-09-13T20:28:31.338867Z"van Luu, Do"https://zbmath.org/authors/?q=ai:do-van-luu."Linh, Pham Thi"https://zbmath.org/authors/?q=ai:linh.pham-thiSummary: Fritz John and Kuhn-Tucker necessary conditions for weak quasi-efficiency of multiobjective fractional optimization problems with equality, inequality and set constraints are derived. Under asumptions on asymptotic pseudoinvexity of the objective and asymptotic quasiinvexity of constraint functions, sufficient conditions for weak quasi-efficiency are also given together with duality theorems of Wolfe and Mond-Weir types.Wolfe type duality for nonsmooth optimization problems with vanishing constraintshttps://zbmath.org/1491.901712022-09-13T20:28:31.338867Z"Ghobadzadeh, Marjan"https://zbmath.org/authors/?q=ai:ghobadzadeh.marjan"Kanzi, Nader"https://zbmath.org/authors/?q=ai:kanzi.nader"Fallahi, Kamal"https://zbmath.org/authors/?q=ai:fallahi.kamalSummary: In this paper, we formulate and study the duality problems in Wolfe type for the mathematical programs with vanishing constraints in nonsmooth case, whereas \textit{S. K. Mishra} et al. [Ann. Oper. Res. 243, No. 1--2, 249--272 (2016; Zbl 1348.90623)] investigated it in smooth case. Also, we derive the weak, strong, strict converse duality results for the problems with Lipschitzian data utilizing Clarke subdifferential.A new Abadie-type constraint qualification for general optimization problemshttps://zbmath.org/1491.901802022-09-13T20:28:31.338867Z"Hejazi, M. Alavi"https://zbmath.org/authors/?q=ai:hejazi.mansoureh-alavi"Movahedian, N."https://zbmath.org/authors/?q=ai:movahedian.nooshinSummary: A non-Lipschitz version of the Abadie constraint qualification is introduced for a nonsmooth and nonconvex general optimization problem. The relationship between the new Abadie-type constraint qualification and the local error bound property is clarified. Also, a necessary optimality condition, based on the pseudo-Jacobians, is derived under the Abadie constraint qualification. Moreover, some examples are given to illustrate the obtained results.Control problems with vanishing Lie bracket arising from complete odd circulant evolutionary gameshttps://zbmath.org/1491.910242022-09-13T20:28:31.338867Z"Griffin, Christopher"https://zbmath.org/authors/?q=ai:griffin.christopher"Fan, James"https://zbmath.org/authors/?q=ai:fan.james-m-robinsSummary: We study an optimal control problem arising from a generalization of rock-paper-scissors in which the number of strategies may be selected from any positive odd number greater than \(1\) and in which the payoff to the winner is controlled by a control variable \(\gamma\). Using the replicator dynamics as the equations of motion, we show that a quasi-linearization of the problem admits a special optimal control form in which explicit dynamics for the controller can be identified. We show that all optimal controls must satisfy a specific second order differential equation parameterized by the number of strategies in the game. We show that as the number of strategies increases, a limiting case admits a closed form for the open-loop optimal control. In performing our analysis we show necessary conditions on an optimal control problem that allow this analytic approach to function.An optimal control problem applied to plasmid-mediated antibiotic resistancehttps://zbmath.org/1491.920722022-09-13T20:28:31.338867Z"Ibargüen-Mondragón, Eduardo"https://zbmath.org/authors/?q=ai:ibarguen-mondragon.eduardo"Esteva, Lourdes"https://zbmath.org/authors/?q=ai:esteva.lourdes"Cerón Gómez, Miller"https://zbmath.org/authors/?q=ai:ceron-gomez.miller(no abstract)