Domain Decomposition Methods in Optimal Control of Partial Differential Equations

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This monograph addresses optimal control issues for elliptic and hyperbolic partial differential equations on networked domains. It aims to describe, develop, and analyze iterative space and time domain decompositions at the infinite-dimensional level. Although domain decomposition methods have a rich history spanning over a century, their significance in numerical analysis of partial differential equations has surged in the last decade, with an emphasis on parallelism. This evolution is exemplified by the recent 15th annual conference focused on this topic. While not exhaustive, the monograph references works like that of Quarteroni and Valli as contemporary sources on domain decomposition methods. The focus here is to contextualize these methods within virtual optimal control problems, treating optimal control challenges for partial differential equations and their decompositions through an all-at-once approach. This approach prioritizes decomposition techniques that can be viewed as virtual optimal control problems, which, along with the actual control problem from a practical application, generate a series of individual optimal control problems on the subdomains that are iteratively decoupled across interfaces.