Stochastic Optimal Control with Delay in the Control I: solution through partial smoothing.

Stochastic optimal control problems governed by delay equations with delay in the control are usually more dicult to study than those in which the delay appears only in the state. This is particularly true when we look at the associated Hamilton{Jacobi{Bellman (HJB) equation. Indeed, even in the simplied setting (introduced rst by Vinter and Kwong [SIAM J. Control Optim., 19 (1981), pp. 139{153] for the deterministic case) the HJB equation is an innite-dimensional second order semilinear partial dierential equation that does not satisfy the so-called structure condition, which substantially means that the control can act on the system, modifying its dynamics at most along the same directions upon which the noise acts. The absence of such a condition|together with the lack of smoothing properties, a common feature of problems with delay|prevents the use of known techniques (based on backward stochastic dierential equations or on the smoothing properties of the linear part) to prove the existence of regular solutions of this HJB equation, and so no results in this direction have been proved till now. In this paper we provide a result on existence of regular solutions of HJB equations of this type. This opens the door to proving existence of optimal feedback controls, a task that will be accomplished in the companion paper [F. Gozzi and F. Masiero, SIAM J. Control Optim., 55 (2017), pp. 3013{3038]. The main tool used is a partial smoothing property that we prove for the transition semigroup associated to the uncontrolled problem. Such results hold for a specic class of equations and data which arise naturally in many applied problems.