Optimal control elliptic systems with distributed and boundary controls

Abstract

In this paper, we consider a nonlinear elliptic control system with distributed and boundary controls and with the integral cost functional. Some sufficient conditions under which the solutions of the elliptic systems continuously depend on boundary and distributed controls are presented. The proof of the result is based on variational methods. These conditions are used to obtain some existence theorem for the optimal control problem governed by nonlinear elliptic equations with distributed and boundary controls. In the final part, we consider the question of stability of the optimal controls and trajectories. Applying some results of the Γ -convergence theory, we prove sufficient conditions under which small variations of parameters of given system imply small changes of optimal processes.