Necessary Optimality Conditions for Control Problems and the Stone--Cech Compactification

Abstract

This paper deals with optimal control problems of parabolic equations in the presence of pointwise state constraints. We consider bounded controls which act in the initial condition of the state equation. The state variable is a bounded continuous function on the domain $Q=\Omega\times ]0,T[$ but is not continuous on $\overline Q$. In this case, the multiplier associated with state constraints is a regular bounded finitely additive measure on Q (but not a $\sigma$-additive one). Using some properties of the Stone--\u{C}ech compactification, we prove a decomposition theorem for this measure which allows us to interpret the adjoint equation in a classical sense. We obtain new optimality conditions for these kinds of problems, and we apply these results to the case of bilateral constraints.