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Optimal Control Problems for Partial Differential Equations on Reticulated Domains
After over 50 years of increasing scientific interest, optimal control of partial differential equations (PDEs) has developed into a well-established discipline in mathematics with myriad applications to science and engineering. As the field has grown, so too has the complexity of the systems it describes; the numerical realization of optimal controls has become increasingly difficult, demanding ever more sophisticated mathematical tools. A comprehensive monograph on the subject, Optimal Control of Partial Differential Equations on Reticulated Domains is intended to address some of the obstacles that face researchers today, particularly with regard to multi-scale engineering applications involving hierarchies of grid-like domains. Bringing original results together with others previously scattered across the literature, it tackles computational challenges by exploiting asymptotic analysis and harnessing differences between optimal control problems and their underlying PDEs. The book consists of two parts, the first of which can be viewed as a compendium of modern optimal control theory in Banach spaces. The second part is a focused, in-depth, and self-contained study of the asymptotics of optimal control problems related to reticulated domains—the first such study in the literature. Specific topics covered in the work include: * a mostly self-contained mathematical theory of PDEs on reticulated domains; * the concept of optimal control problems for PDEs in varying such domains, and hence, in varying Banach spaces; * convergence of optimal control problems in variable spaces; * an introduction to the asymptotic analysis of optimal control problems; * optimal control problems dealing with ill-posed objects on thin periodic structures, thick periodic singular graphs, thick multi-structures with Dirichlet and Neumann boundary controls, and coefficients on reticulated structures. Serving as both a text on abstract optimal control problems and a monograph where specific applications are explored, this book is an excellent reference for graduate students, researchers, and practitioners in mathematics and areas of engineering involving reticulated domains.
Introduction -- Part I. Asymptotic Analysis of Optimal Control Problems for Partial Differential Equations: Basic Tools -- Background Material on Asymptotic Analysis of External Problems -- Variational Methods of Optimal Control Theory -- Suboptimal and Approximate Solutions to External Problems -- Introduction to the Asymptotic Analysis of Optimal Control Problems: A Parade of Examples -- Convergence Concepts in Variable Banach Spaces -- Convergence of Sets in Variable Spaces -- Passing to the Limit in Constrained Minimization Problems -- Part II. Optimal Control Problems on Periodic Reticulated Domains: Asymptotic Analysis and Approximate Solutions -- Suboptimal Control of Linear Steady-States Processes on Thin Periodic Structures with Mixed Boundary Controls -- Approximate Solutions of Optimal Control Problems for Ill-Posed Objects on Thin Periodic Structures -- Asymptotic Analysis of Optimal Control Problems on Periodic Singular Structures -- Suboptimal Boundary Control of Elliptic Equations in Domains with Small Holes -- Asymptotic Analysis of Elliptic Optimal Control Problems in Thick Multi-Structures with Dirichlet and Neumann Boundary Controls -- Gap Phenomenon in Modeling of Suboptimal Controls to Parabolic Optimal Control Problems in Thick Multi-Structures -- Boundary Velocity Suboptimal Control of Incompressible Flow in Cylindrically Perforated Domains -- Optimal Control Problems in Coefficients: Sensitivity Analysis and Approximation -- References -- Index.
Calculus of Variations and Optimal Control; Optimization
Appl.Mathematics/Computational Methods of Engineering
Systems & Control: Foundations & Applications