Systems Research Institute of the Polish Academy of Sciences
Abstract
A problem of sparse optimal boundary control for a semilinear parabolic partial differential equation is considered, where pointwise bounds on the control and mixed pointwise control-state constraints are given. A standard quadratic objective functional is to be minimized that includes a Tikhonov regularization term and the L1-norm of the control accounting for the sparsity. Applying a recent linearization theorem, we derive first-order necessary optimality conditions in terms of a variational inequality under linearized mixed control state constraints. Based on this preliminary result, a Lagrange multiplier rule with bounded and measurable multipliers is derived and sparsity results on the optimal control are demonstrated.The first author was partially supported by the Spanish Ministerio de Economía y Competitividad under project MTM2017-83185-P. The second author was supported by the collaborative research center SFB 910, TU Berlin, project B6
