On the Variational Stability of a Class of Nonlinear Parabolic Optimal Control Problems

Abstract

In this paper we study parametric optimal control problems governed by a non-linear parabolic equation in divergence form. The parameter appears in all the data of the problem, including the partial differential operator. Using as tools the G-convergence of operators and the $\Gamma$-convergence of functionals, we show that the set-valued map of optimal pairs is upper-semicontinuous with respect to the parameter and the optimal value function responds continuously to changes of the parameter. Finally, in the case of semilinear systems we show that our framework can also incorporate systems with weakly convergent coefficients.