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A class of optimal control problems governed by semilinear parabolicequations with mixed pointwise are considered. We give some criteria underwhich Lagrange multipliers belong to $L^p$-spaces. We then establish first andsecond-order optimality conditions of KKT-type for the problem. Moreover, weshow that if the initial value is good enough and boundary $\partial\Omega$ hasa property of positive geometric density, then multipliers and optimalsolutions are H\"{o}lder continuous.

Regularity of multipliers and optimal solutions in second-order optimality conditions for semilinear parabolic optimal control problems