HJB Equations and Stochastic Control on Half-Spaces of Hilbert Spaces
将HJB方程在希尔伯特空间中的温和解理论扩展到半空间域，证明了半线性HJB方程解的存在唯一性，并应用于退出时间最优控制问题，得到值函数的正则性和反馈控制形式。
Abstract In this paper, we study a first extension of the theory of mild solutions for Hamilton–Jacobi–Bellman (HJB) equations in Hilbert spaces to the case where the domain is not the whole space. More precisely, we consider a half-space as domain, and a semilinear HJB equation. Our main goal is to establish the existence and the uniqueness of solutions to such HJB equations, which 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.