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 x (0, T) \subset \BbbR n+1, where the control is assumed to be in 0;,0 (Q)]*, rather than in L2(Q), which is more common. While the latter ensures 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 completely unstructured but shape regular simplicial meshes, we derive a priori estimates for the error II uwidetilde \\varrho h - uIIL2(Q) between the computed space-time finite element solution uwidetilde \\varrho h and the target function 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 u. These estimates lead to the optimal choice \varrho = h2 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.