Space-Time Finite Element Methods for Distributed Optimal Control of the Wave Equation

SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 452-475, February 2024.
Abstract. We consider space-time tracking-type distributed optimal control problems for the wave equation in the space-time domain [math], where the control is assumed to be in the energy space [math], rather than in [math], which is more common. While the latter ensures a unique state in the Sobolev space [math], this does not define a solution isomorphism. Hence, we use an appropriate state space [math] such that the wave operator becomes an isomorphism from [math] onto [math]. Using space-time finite element spaces of piecewise linear continuous basis functions on completely unstructured but shape regular simplicial meshes, we derive a priori estimates for the error [math] between the computed space-time finite element solution [math] and the target function [math] with respect to the regularization parameter [math], and the space-time finite element mesh size [math], depending on the regularity of the desired state [math]. These estimates lead to the optimal choice [math] in order to define the regularization parameter [math] for a given space-time finite element mesh size [math] or to determine the required mesh size [math] when [math] is a given constant representing the costs of the control. The theoretical results will be supported by numerical examples with targets of different regularities, including discontinuous targets. Furthermore, an adaptive space-time finite element scheme is proposed and numerically analyzed.