Viscosity Solutions to First Order Path-Dependent Hamilton-Jacobi-Bellman Equations in Hilbert Space

TL;DR

This paper introduces a new notion of viscosity solutions for first order path-dependent Hamilton-Jacobi-Bellman equations in Hilbert spaces, establishing their uniqueness and consistency with classical solutions for optimal control problems.

Contribution

It defines and analyzes viscosity solutions for path-dependent HJB equations in infinite-dimensional Hilbert spaces, linking them to optimal control problem value functionals.

Findings

Proves the uniqueness of viscosity solutions for the PHJB equations.

Shows the consistency of viscosity solutions with classical solutions.

Establishes stability properties of the solutions.

Abstract

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. We also show that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions, and satisfies a stability property.