Abstract
In this article, we combine a domain decomposition method in space and time for optimal control problems with PDE-constraints described in [2] to a simultaneous space-time decomposition applied to optimal control problems for systems of linear hyperbolic equations with distributed control. We thereby extend the recent work [31, 32] and answer a long standing open question as to whether the combination of time- and space-domain decomposition for the method under consideration can be put into one single convergent iteration procedure. The algorithm is designed for a semi-elliptic system of equations obtained from the hyperbolic optimality system by the way of reduction to the adjoint state. The focus is on the relation to the classical procedure introduced by P. L. Lions [25] for elliptic problems.
Highlights
Spatial domain decomposition methods as well as time-domain decomposition for partial differential equations (PDEs) have been a subject of intense research in the past
The wealth of articles and even monographs related to the topic, shrinks drastically when it comes to the decomposition of optimal control problems for partial differential equations and even more when convergence of the iterative methods is considered on the PDE-level
Our particular interest is in such methods which result in a decomposed system which, in turn, can be seen as an optimality system associated with a virtual control problem on a smaller space-time domain
Summary
Spatial domain decomposition methods as well as time-domain decomposition for partial differential equations (PDEs) have been a subject of intense research in the past. Our particular interest is in such methods which result in a decomposed system which, in turn, can be seen as an optimality system associated with a virtual control problem on a smaller space-time domain In this sense, one aims at the decomposition of the original. These methods, which consist of a coupling of coarse grain discrete-in-time solutions at the break points with a parallel computation of full (respectively, small grain) solutions on the subintervals, were first developed for the mere simulation of nonlinear PDEs. In the article [4], the authors, for the first time, considered the time-domain decomposition of optimal control problems for the time-dependent Maxwell system.
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