Sparse optimal control for a semilinear heat equation with mixed control-state constraints – regularity of Lagrange multipliers

An optimal control problem for a semilinear heat equation with distributed control is discussed, where two-sided pointwise box constraints on the control and two-sided pointwise mixed control-state constraints are given. The objective functional is the sum of a standard quadratic tracking type part and a multiple of the L 1 -norm of the control that accounts for sparsity. Under a certain structural condition on almost active sets of the optimal solution, the existence of integrable Lagrange multipliers is proved for all inequality constraints. For this purpose, a theorem by Yosida and Hewitt is used. It is shown that the structural condition is fulfilled for all sufficiently large sparsity parameters. The sparsity of the optimal control is investigated. Eventually, higher smoothness of Lagrange multipliers is shown up to Hölder regularity.