Optimal Control Problems with Mixed Control-State Constraints

\noindent We consider control problems governed by semilinear parabolic equations in the presence of pointwise mixed control-state constraints. We obtain optimality conditions with finitely additive measures as multipliers associated to the mixed constraints. We study the regularity of the multipliers for different problems. In particular, when a monotonicity or a separation condition is satisfied, we prove that multipliers are bounded measurable functions.