Domain Decomposition Methods in Optimal Control of Partial Differential Equations

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.

1 Introduction.- 2 Background Material on Domain Decomposition.- 2.1 Introduction.- 2.2 Domain Decomposition for 1-d Problems.- 2.2.1 Unbounded Domains.- 2.2.2 Bounded Domains.- 2.2.3 Semi-discretization.- 2.3 Domain Decomposition Methods for Elliptic Problems.- 2.3.1 Review of Basic Methods.- 2.3.2 Virtual Controls.- 2.3.3 The Basic Algorithm of P.-L. Lions.- 2.3.4 An Augmented Lagrangian Formulation.- 2.3.5 General Elliptic Problems and More General Splittings.- 2.3.6 An a Posteriori Error Estimate.- 2.3.7 Interpretation as a Damped Richardson iteration.- 2.3.8 A Serial One-Dimensional Problem.- 3 Partial Differential Equations on Graphs.- 3.1 Introduction.- 3.2 Partial Differential Operators on Graphs.- 3.3 Elliptic Problems on Graphs.- 3.3.2 Domain Decomposition.- 3.3.3 Convergence.- 3.3.4 Interpretation as a Richardson Iteration.- 3.4 Hyperbolic Problems on Graphs.- 3.4.1 The Model.- 3.4.2 The Domain Decomposition Procedure.- 4 Optimal Control of Elliptic Problems.- 4.1 Introduction.- 4.2 Distributed Controls.- 4.2.2 Domain Decomposition.- 4.2.3 A Complex Helmholtz Problem and its Decomposition.- 4.2.4 Convergence.- 4.2.5 Methods for Elliptic Optimal Control Problems.- 4.2.6 An A Posteriori Error Estimate.- 4.3 Boundary Controls.- 4.3.2 Domain Decomposition.- 4.3.3 Convergence.- 4.3.4 An A Posteriori Error Estimate.- 5 Control of Partial Differential Equations on Graphs.- 5.1 Introduction.- 5.2 Elliptic Problems.- 5.2.1 The Global Optimal Control Problem on a Graph.- 5.2.2 Domain Decomposition.- 5.2.3 Distributed Controls.- 5.2.4 Boundary Controls.- 5.3 Hyperbolic Problems.- 5.3.1 The Global Optimal Control Problem on a Graph.- 5.3.2 The Domain Decomposition Procedure.- 6 Control of Dissipative Wave Equations.- 6.1 Introduction.- 6.2 Optimal Dissipative Boundary Control.- 6.2.1 Setting the Problem.- 6.2.2 Existence and Regularity of Solutions.- 6.2.3 The Global Optimality System.- 6.3 Time Domain Decomposition.- 6.3.1 Description of the Algorithm.- 6.3.2 Convergence of the Iterates.- 6.3.3 A Posteriori Error Estimates.- 6.3.4 Extension to General Dissipative Control Systems.- 6.4 Decomposition of the Spatial Domain.- 6.4.1 Description of the Algorithm.- 6.4.2 Convergence of the Iterates.- 6.4.3 A Posteriori Error Estimates.- 6.5 Space and Time Domain Decomposition.- 6.5.1 Sequential Space-Time Domain Decomposition.- 6.5.2 Sequential Time-Space Domain Decomposition.- 7 Boundary Control of Maxwell’s System.- 7.1 Introduction.- 7.2 Optimal Dissipative Boundary Control.- 7.2.1 Setting the Problem.- 7.2.2 Existence and Uniqueness of Solution.- 7.2.3 The Global Optimality System.- 7.3 Time Domain Decomposition.- 7.3.1 Description of the Algorithm.- 7.3.2 Convergence of the Iterates.- 7.3.3 A Posteriori Error Estimates.- 7.4 Decomposition of the Spatial Domain.- 7.4.1 Description of the Algorithm.- 7.4.2 Convergence of the Iterates.- 7.4.3 A Posteriori Error Estimates.- 7.5 Time and Space Domain Decomposition.- 7.5.1 Sequential Space-Time Domain Decomposition.- 7.5.2 Sequential Time-Space Domain Decomposition.- 8 Control of Conservative Wave Equations.- 8.1 Introduction.- 8.2 Optimal Boundary Control.- 8.2.1 Setting the Problem.- 8.2.2 Existence and Regularity of Solutions.- 8.2.3 The Global Optimality System.- 8.3 Time Domain Decomposition.- 8.3.1 Description of the Algorithm.- 8.3.2 Convergence of the Iterates.- 8.3.3 A Posteriori Error Estimates.- 8.3.4 Extension to General Conservative Control Systems.- 8.4 Decomposition of the Spatial Domain.- 8.4.1 The Local Optimality Systems.- 8.4.2 The Domain Decomposition Algorithm.- 8.4.3 Convergence of the Iterates.- 8.5 The Exact Reachability Problem.- 8.5.1 The Global Optimality System.- 8.5.2 The Limit of the Local Optimality Systems.- 8.5.3 Application to Domain Decomposition.- 8.5.4 Convergence to the Global Optimality System.- 9 Domain Decomposition for 2-D Networks.- 9.1 Elliptic Systems on 2-D Networks.- 9.1.2 Examples.- 9.1.3 Existence and Uniqueness of Solutions.- 9.1.4 Domain Decomposition.- 9.1.5 Convergence of the Algorithm.- 9.2 Optimal Control on 2-D Networks.- 9.2.1 Optimal Final Value Control.- 9.2.2 Existence and Regularity of Solutions.- 9.3 Decomposition of the Spatial Domain.- 9.3.2 The Decomposition Algorithm.- 9.3.3 Convergence of the Algorithm.