Space-time finite element methods for distributed optimal control of the wave equation
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We consider space-time tracking type distributed optimal control problems for the wave equation in the space-time domain Q:= Ω× (0,T) ⊂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 u_ϱ h-u_L^2(Q) between the computed space-time finite element solution u_ϱ h and the target function u with respect to the regularization parameter ϱ, 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 ϱ=h^2 in order to define the regularization parameter ϱ for a given space-time finite element mesh size h, or to determine the required mesh size h when ϱ 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.
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