Necessary Conditions for Optimal Controls of Semilinear Parabolic Equations with Control in the Leading Term

Abstract

An optimal control problem for seconde order semilinear parabolic partial differential equation is considered. The equation is in divergence form with the leading term containing the control. By the method of homogenizing spike variation, necessary conditions for optimal controls are established. The key to such a method is to modify the usual spike variational technique by taking into account the homogenization techniques for parabolic equations, together with some idea from the theory of relaxed controls. The method we used here can also treat problems with state constraints by adding some well‐known penalty arguments involving the application of Ekeland’s variational principle and finite co‐dimensionality of certain sets.