On the Existence of Optimal Controls

The optimal control problem where the state is governed by an Itô stochastic differential equation (possibly just an ordinary differential equation) is formulated in martingale terms. Under a coercivity condition (which is weaker than compactness of the control set), a convexity condition, and mild continuity hypotheses on the data, it is shown by the direct method that optimal controls exist. Hard and soft constraints are allowed. In the absence of soft constraints it is shown that there exists an optimal control that is a function only of the present time and state, i.e., the synthesis problem has a solution. The main tool here is Krylov’s Markovian Selection Theorem.