2022/11/04 by Richard Löscher, Olaf Steinbach, Löscher, Richard +1 · 1 citation
Engineering · Mathematics · #35L05 #49M41 #65M15 #65M60 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2211.02562
openalex publication_date 2022/11/04 · openalex created_date 2022/11/12 · openalex updated_date 2026/07/28
We consider space-time tracking type distributed optimal control problems for the wave equation in the space-time domain Q:= Ω× (0,T) ⊂ ℝn+1, where the control is assumed to be in the energy space [H0;,01,1(Q)]^*, rather than in L2(Q) which is more common. While the latter ensures a unique state in the Sobolev space H1,10;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 [H0;,01,1(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 ‖\widetildeu\varrho h-u‖L2(Q) between the computed space-time finite element solution \widetildeu\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.