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

TL;DR

This paper develops a novel approach using partial smoothing properties to establish the existence of regular solutions for complex stochastic control problems with delays in the control, enabling the derivation of optimal feedback controls.

Contribution

It introduces a new method based on partial smoothing to solve HJB equations with delays in control, which were previously intractable due to lack of smoothing properties.

Findings

Proves partial smoothing property for the transition semigroup.

Establishes existence of regular solutions for delayed control HJB equations.

Derives optimal feedback controls for problems with pointwise delay.

Abstract

Stochastic optimal control problems governed by delay equations with delay in the control are usually more difficult to study than the 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 semilinear Partial Differential Equation (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 the known techniques (based on Backward Stochastic Differential Equations (BSDEs) or on the smoothing properties of the linear part) to prove the existence of regular solutions of this HJB equation and so no results on this direction have been proved till now. In this paper we provide a result on existence of regular solutions of such kind of HJB equations and we use it to solve completely the corresponding control problem finding optimal feedback controls also in the more difficult case of pointwise delay. The main tool used is a partial smoothing property that we prove for the transition semigroup associated to the uncontrolled problem. Such results holds for a specific class of equations and data which arises naturally in many applied problems.