Space-time finite element methods for distributed optimal control of the wave equation

We consider space-time tracking type distributed optimal control problems for the wave equation in the space-time domain $Q:= \Omega \times (0,T) \subset {\mathbb{R}}^{n+1}$, where the control is assumed to be in the energy space $[H_{0;,0}^{1,1}(Q)]^*$, rather than in $L^2(Q)$ which is more common. While the latter ensures a unique state in the Sobolev space $H^{1,1}_{0;0,}(Q)$, this does not define a solution isomorphism. Hence we use an appropriate state space $X$ such that the wave operator becomes an isomorphism from $X$ onto $[H_{0;,0}^{1,1}(Q)]^*$. 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 $\|\widetilde{u}_{\varrho h}-\overline{u}\|_{L^2(Q)}$ between the computed space-time finite element solution $\widetilde{u}_{\varrho h}$ and the target function $\overline{u}$ with respect to the regularization parameter $\varrho$, and the space-time finite element mesh-size $h$, depending on the regularity of the desired state $\overline{u}$. These estimates lead to the optimal choice $\varrho=h^2$ in order to define the regularization parameter $\varrho$ for a given space-time finite element mesh size $h$, or to determine the required mesh size $h$ when $\varrho$ 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.