Stochastic Control Problems with Unbounded Control Operators: solutions through generalized derivatives

TL;DR

This paper addresses stochastic control problems in infinite-dimensional spaces with unbounded control operators, introducing generalized derivatives to solve the associated HJB equations and derive feedback controls.

Contribution

It develops a novel concept of partial derivatives tailored for unbounded control operators, enabling the proof of regular solutions to HJB equations in complex stochastic control settings.

Findings

Established a new method for solving HJB equations with unbounded control operators.

Proved the existence of regular solutions allowing feedback control synthesis.

Applied the approach to boundary control and delay equations in Hilbert spaces.

Abstract

This paper deals with a family of stochastic control problems in Hilbert spaces which arises in typical applications (such as boundary control and control of delay equations with delay in the control) and for which is difficult to apply the dynamic programming approach due to the unboudedness of the control operator and to the lack of regularity of the underlying transition semigroup. We introduce a specific concept of partial derivative, designed for this situation, and we develop a method to prove that the associated HJB equation has a solution with enough regularity to find optimal controls in feedback form.