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.

Emphasis 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 decompositions by an all-at-once approach Development of a general theory for systems of partial differential equations on multi-linked or networked domains

Inhalt
1 Introduction.- 2 Background Material on Domain Decomposition.- 2.1 Introduction.- 2.2 Domain Decomposition for 1-d Problems.- 2.3 Domain Decomposition Methods for Elliptic Problems.- 3 Partial Differential Equations on Graphs.- 3.1 Introduction.- 3.2 Partial Differential Operators on Graphs.- 3.3 Elliptic Problems on Graphs.- 3.4 Hyperbolic Problems on Graphs.- 4 Optimal Control of Elliptic Problems.- 4.1 Introduction.- 4.2 Distributed Controls.- 4.3 Boundary Controls.- 5 Control of Partial Differential Equations on Graphs.- 5.1 Introduction.- 5.2 Elliptic Problems.- 5.3 Hyperbolic Problems.- 6 Control of Dissipative Wave Equations.- 6.1 Introduction.- 6.2 Optimal Dissipative Boundary Control.- 6.3 Time Domain Decomposition.- 6.4 Decomposition of the Spatial Domain.- 6.5 Space and Time Domain Decomposition.- 7 Boundary Control of Maxwell's System.- 7.1 Introduction.- 7.2 Optimal Dissipative Boundary Control.- 7.3 Time Domain Decomposition.- 7.4 Decomposition of the Spatial Domain.- 7.5 Time and Space Domain Decomposition.- 8 Control of Conservative Wave Equations.- 8.1 Introduction.- 8.2 Optimal Boundary Control.- 8.3 Time Domain Decomposition.- 8.4 Decomposition of the Spatial Domain.- 8.5 The Exact Reachability Problem.- 9 Domain Decomposition for 2-D Networks.- 9.1 Elliptic Systems on 2-D Networks.- 9.2 Optimal Control on 2-D Networks.- 9.3 Decomposition of the Spatial Domain.