  In this paper we study a first extension of the theory of mild solutions for
HJB equations in Hilbert spaces to the case when the domain is not the whole
space. More precisely, we consider a half-space as domain, and a semilinear
Hamilton-Jacobi-Bellman (HJB) equation. Our main goal is to establish the
existence and the uniqueness of solutions to such HJB equations, that are
continuously differentiable in the space variable. We also provide an
application of our results to an exit time optimal control problem and we show
that the corresponding value function is the unique solution to a semilinear
HJB equation, possessing sufficient regularity to express the optimal control
in feedback form. Finally, we give an illustrative example.
