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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.

Local quadratic convergence of SQP for elliptic optimal control problems with mixed control-state constraints

Griesse R., Metla N., Rosch A.

Control and Cybernetics

2010

Vol. 39, no 3

717-738

Semilinear elliptic optimal control problems with pointwise control and mixed control-state constraints are considered. Necessary and sufficient optimality conditions are given. The equivalence of the SQP method and Newton's method for a generalized equation is discussed. Local quadratic convergence of the SQP method is proved.