Lipschitz Stability of Solutions to Parametric Optimal Control Problems for Parabolic Equations

Abstract

A class of parametric optimal control problems for semilinear parabolic equations is considered. Using recent regularity results for solutions of such equations, sufficient condi-tions are derived under which the solutions to optimal control problems are locally Lipschitz continuous functions of the parameter in the $L^{\infty }$-norm. It is shown that these conditions are also necessary, provided that the dependence of data on the parameter is sufficiently strong.