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

Alessandro Calvia (), Gianluca Cappa (), Fausto Gozzi () and Enrico Priola ()
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Alessandro Calvia: Università di Parma
Gianluca Cappa: LUISS University
Fausto Gozzi: LUISS University
Enrico Priola: University of Pavia

Journal of Optimization Theory and Applications, 2023, vol. 198, issue 2, No 10, 710-744

Abstract: 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.

Keywords: Stochastic control; Second-order Hamilton–Jacobi–Bellman equations in infinite dimension; Regular solutions; Nonlinear Partial Differential Equations in domains; Smoothing properties of transition semigroups; 35R15 (PDEs on infinite-dimensional (e.g.; function) spaces); 47D07 (Markov semigroups and applications to diffusion processes); 49L12 (Hamilton–Jacobi equations in optimal control and differential games); 49L20 (Dynamic programming in optimal control and differential games); 93E20 (Optimal stochastic control) (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10957-023-02208-1

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