Minimax Control of Parabolic Systems with State Constraints

Abstract

In this paper we study a minimax control problem for parabolic equations in the presence of pointwise state constraints. The terminology minimax here refers to a cost functional defined with a $L^{\infty}$-norm. The directional derivatives of the $L^{\infty}$-norm are elements of $(L^{\infty})'$. Therefore, the adjoint equation may involve finitely additive measures in place of Radon measures. To overcome this difficulty, we introduce a compactification (of Stone--\u{C}ech type). We prove necessary optimality conditions which are new, both in the case with no state constraints and in the case with state constraints. Under some convexity conditions, these optimality conditions are also sufficient.