| Contents |
| This talk deals with optimal control problems for instationary wave equations involving a sparsity functional as control cost term. In particular the norms of the Bochner space $L^2((0,T),\mathcal M(\Omega))$ or of the space of vector measures $\mathcal M(\Omega,L^2(0,T))$ are chosen to induce sparsity of the optimal control in space direction and to have $L^2$-regularity in time. One important difference between these two approaches is that the first one allows for a time dependent support of the optimal measure, e.g., a moving point source, whereas the second only allows for a time independent support of the optimal measure. Further differences regarding the well posedness of the state equation and the first order optimality conditions are discussed. The sparsity properties of the optimal controls are derived from these first order optimality conditions. A regularized version of the problem can be solved by a semi-smooth Newton-method. The talk is concluded with some numerical examples. |
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