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

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.