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Optimal sparse boundary control for a semilinear parabolic equation with mixed control-state constraints

Casas Eduardo, Troltzsch Fredi

Control and Cybernetics

2019

Vol. 48, No. 1

89--124

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.