Locally Lipschitz stability of solutions to a parametric parabolic optimal control problem with mixed pointwise constraints

A class of parametric optimal control problems governed by semilinear parabolic equations with mixed pointwise constraints is investigated. The perturbations appear in the objective functional, the state equation and in mixed pointwise constraints. By analyzing regularity and establishing stability condition of Lagrange multipliers we prove that, if the strictly second-order sufficient condition for the unperturbed problem is valid, then the solutions of the problems as well as the associated Lagrange multipliers are locally Lipschitz continuous functions of parameters.