Regular Lagrange Multipliers for Control Problems with Mixed Pointwise Control-State Constraints

A class of quadratic optimization problems in Hilbert spaces is considered, where pointwise box constraints and constraints of bottleneck type are given. The main focus is to prove the existence of regular Lagrange multipliers in L-spaces. This question is solved by investigating the solvability of a Lagrange dual quadratic problem. The theory is applied to different optimal control problems for elliptic and parabolic partial differential equations with mixed pointwise control-state constraints.