Regularity of multipliers and second-order optimality conditions of KKT-type for semilinear parabolic control problems

A class of optimal control problems governed by semilinear parabolic equations with mixed constraints and a box constraint for control variable is considered. We show that if the separation condition is satisfied, then both optimality conditions of KKT-type and regularity of multipliers are fulfilled. Moreover, we show that if the initial value is good enough and boundary $\mathrm{\partial }\mathrm{\Omega }$ has a property of positive geometric density, then multipliers and optimal solutions are Hölder continuous.