Viscosity solutions to first order path-dependent Hamilton-Jacobi-Bellman equations in Hilbert space

In this article, a notion of viscosity solutions is introduced for first order path-dependent Hamilton-Jacobi-Bellman (PHJB) equations associated with optimal control problems for path-dependent evolution equations in Hilbert space. We identify the value functional of optimal control problems as unique viscosity solution to the associated PHJB equations without the assumption (A.4) on page 231 of Li and Yong (1995). We also show that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions, and satisfies a stability property. (C) 2022 Elsevier Ltd. All rights reserved.

Automated summary

This article introduces the concept of viscosity solutions for first-order path-dependent Hamilton-Jacobi-Bellman (PHJB) equations associated with optimal control problems for path-dependent evolution equations in Hilbert space. We identify the value functional of optimal control problems as the unique viscosity solution to the associated PHJB equations without a specific assumption. We also demonstrate that our notion of viscosity solutions is consistent with classical solutions and exhibits a stability property.