Domain Decomposition Methods in Optimal Control of Partial Differential Equations (International Series of Numerical Mathematics, 148)

This monograph considers problems of optimal control for partial differential equa­ tions of elliptic and, more importantly, of hyperbolic types on networked domains. The main goal is to describe, develop and analyze iterative space and time domain decompositions of such problems on the infinite-dimensional level. While domain decomposition methods have a long history dating back well over one hundred years, it is only during the last decade that they have become a major tool in numerical analysis of partial differential equations. A keyword in this context is parallelism. This development is perhaps best illustrated by the fact that we just encountered the 15th annual conference precisely on this topic. Without attempting to provide a complete list of introductory references let us just mention the monograph by Quarteroni and Valli [91] as a general up-to-date reference on domain decomposition methods for partial differential equations. The emphasis of this monograph is to put domain decomposition methods in the context of so-called virtual optimal control problems and, more importantly, to treat optimal control problems for partial differential equations and their decom­ positions by an all-at-once approach. This means that we are mainly interested in decomposition techniques which can be interpreted as virtual optimal control problems and which, together with the real control problem coming from an un­ derlying application, lead to a sequence of individual optimal control problems on the subdomains that are iteratively decoupled across the interfaces. Front Matter....Pages i-xiii Introduction....Pages 1-7 Background Material on Domain Decomposition....Pages 9-69 Partial Differential Equations on Graphs....Pages 71-106 Domain Decomposition for Elliptic Optimal Control Problems....Pages 107-129 Optimal Control of One-Dimensional Partial Differential Equations on Graphs....Pages 131-157 Domain Decomposition in Optimal Final Value Control of Dissipative Wave Equations....Pages 159-256 Domain Decomposition in Optimal Final Value Boundary Control of Maxwell’s System....Pages 257-320 Optimal Final Value Boundary Control of Conservative Wave Equations....Pages 321-374 Domain Decomposition for Distributed Parameter Systems on 2-D Networks....Pages 375-433 Back Matter....Pages 435-446 While domain decomposition methods have a long history dating back well over one hundred years, it is only during the last decade that they have become a major tool in numerical analysis of partial differential equations. This monograph considers problems of optimal control for partial differential equations of elliptic and, more importantly, of hyperbolic types on networked domains. The main goal is to describe, develop and analyze iterative space and time domain decompositions of such problems on the infinite dimensional level. While domain decomposition methods have a long history dating back well over one hundred years, it is only during the last decade that they have become a major tool in numerical analysis of partial differential equations. This monograph emphasizes domain decomposition methods in the context of so-called virtual optimal control problems and treats optimal control problems for partial differential equations and their decompositions using an all-at-once approach. Preface 1. Introduction 2. Background Material on Domain Decomposition 3. Partial Differential Equations on Graphs 4. Optimal Control of Elliptic Problems 5. Control of Partial Differential Equations on Graphs 6. Control of Dissipative Wave Equations 7. Boundary Control of Maxwell's System 8. Control of Conservative Wave Equations 9. Domain Decomposition for 2D-Networks Bibliography Index.