Georgia Institute of Technology College of Sciences

Stochastic optimal control problems governed by delay equations with delay in the control are usually more difficult to study than the ones when 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 simplified setting (introduced first by Vinter and Kwong for the deterministic case) the HJB equation is an infinite dimensional second order semi-linear PDE that does not satisfy the so-called structure condition which substantially means that "the noise enters the system with the control". The absence of such condition, together with the lack of smoothing properties which is a common feature of problems with delay, prevents the use of known techniques (based on Backward Stochastic Differential Equations or on the smoothing properties of the linear part) to prove the existence of regular solutions to this HJB equation and thus no results in this direction have been proved till now. In this talk we will discuss results about existence of regular solutions of this kind of HJB equations and their use in solving the corresponding control problem by finding optimal feedback controls, also in the more difficult case of pointwise delay. This is a joint work with Federica Masiero.