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.

HJB Equations and Stochastic Control on Half-Spaces of Hilbert Spaces / Calvia, Alessandro; Cappa, Gianluca; Gozzi, Fausto; Priola, E.. - In: JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS. - ISSN 0022-3239. - 198:2(2023), pp. 710-744. [10.1007/s10957-023-02208-1]

HJB Equations and Stochastic Control on Half-Spaces of Hilbert Spaces

Calvia A.;Cappa G.;Gozzi F.;
2023

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.
2023
Nonlinear Partial Differential Equations in domains, Regular solutions, Second-order Hamilton–Jacobi–Bellman equations in infinite dimension, Smoothing properties of transition semigroups, Stochastic control
HJB Equations and Stochastic Control on Half-Spaces of Hilbert Spaces / Calvia, Alessandro; Cappa, Gianluca; Gozzi, Fausto; Priola, E.. - In: JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS. - ISSN 0022-3239. - 198:2(2023), pp. 710-744. [10.1007/s10957-023-02208-1]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11385/234943
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