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 of these problems at an infinite-dimensional level. Although domain decomposition methods have a long history, they have gained prominence in the past decade as a vital tool in the numerical analysis of partial differential equations, with parallelism being a key aspect. This evolution is exemplified by the recent 15th annual conference focused on this topic. While not exhaustive, the work references the monograph by Quarteroni and Valli as a contemporary source on domain decomposition methods. The focus here is to contextualize these methods within virtual optimal control problems and to approach optimal control problems for partial differential equations and their decompositions through an all-at-once strategy. This approach emphasizes decomposition techniques interpretable as virtual optimal control problems, which, together with the real control problem from an application, yield a series of individual optimal control problems on subdomains that are iteratively decoupled across interfaces.