Second-order optimality conditions and regularity of Lagrange multipliers for mixed optimal control problems

Abstract

This paper deals with second-order optimality conditions and regularity of Lagrange multipliers for a class of optimal control problems governed by semilinear elliptic equations with mixed pointwise constraints in which controls act both in the domain and on the boundary. We give some criteria under which the optimality conditions are of KKT type and the multipliers are of $$L^p$$ -spaces. Moreover, we show that the multipliers are Lipschitz continuous functions.