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Domain decomposition methods in optimal control of partial differential equations. (English) Zbl 1059.49002

ISNM. International Series of Numerical Mathematics 148. Basel: Birkhäuser (ISBN 3-7643-2194-6/hbk). xiii, 443 p. (2004).
This outstanding monograph is concerned with a domain decomposition techniques in space and time for optimal control problems governed by partial differential equations of elliptic and hyperbolic type on networked domains. The main purpose is to describe, develop and analyze iterative space and time domain decompositions of such problems on the infinite-dimensional level, and introduces readers to the context of so-called virtual optimal control problems, and, more importantly, to treat optimal control problems for partial differential equations and their decomposition by all-at-once approach. The authors give a complete treatment of decomposition techniques which can be interpreted as virtual optimal control problems and which together with the real control problem coming from an underlying application, lead to a sequence of individual optimal control problems on the subdomains that are iteratively decoupled across the interfaces. The decomposition of the optimality systems of the global problem into optimality systems related to iteratively coupled optimal control problems for the subsystems, represents the characteristic attribute of the methods.
The book is divided into two parts. The first part (Chapters 2 through 5) provide the basic ideas, fundamental methods and basic convergence proofs. Here the corresponding sections end with a summary of results in the form of a theorem. Part two is devoted to the study of the problems which are motivated by the material of the earlier chapters. It is mainly directed to specialists and is itself largely self-contained. On the other hand, the Chapters 6 through 9 are organized in a somewhat different style, where results on well-posedness, convergence and a posteriori error estimates are formalized as theorems with subsequent proofs. Precise proofs being usually sufficiently detailed. All results are presented in a systematic and self-contained manner. The abstract results are useful in the study of a wide class of problems arising in the flexible structures consisting of strings, cables, hells, beams and plates that are coupled at mechanical joints. Most of these applications are driven by hyperbolic equations on network-like structures. The conceptual algorithms on an infinite dimensional level, to prove convergence of the algorithms are proposed. The algorithms obtained then provide a mesh-independent framework for numerical realizations and corresponding simulations.
This volume is primarily addressed to applied mathematicians working in the field of partial differential equations and their applications, especially those concerned with optimal control problems aspects. However, the book will also be useful for scientists. from the application areas, in particular, applied scientists from engineering field and physics and graduate level students in advanced computational mechanics, respectively.

MSC:

49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49M27 Decomposition methods
49J20 Existence theories for optimal control problems involving partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65K10 Numerical optimization and variational techniques
35J25 Boundary value problems for second-order elliptic equations
35L10 Second-order hyperbolic equations
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