The optimal control problem for the coefficients of a parabolic equation under phase constraints

Abstract

For the optimal control problem on the coefficients of parabolic equations, we study correctness problems for the setting in the weak topology on the space of controls. We show that in this problem, the objective functional is bounded from below, the set of optimal controls is nonempty, and every minimizing sequence weakly converges to the set of optimal controls. We establish a necessary optimality condition in the form of a generalized Lagrange multiplier rule.