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