Viability for locally monotone evolution inclusions and lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations in infinite dimensions

TL;DR

This paper develops a comprehensive framework for viability and solutions of Hamilton-Jacobi-Bellman equations in infinite-dimensional spaces with locally monotone operators, broadening applicability to complex PDEs and control problems.

Contribution

It generalizes viability and solution results to locally monotone operators, including applications to PDEs and constrained control in infinite dimensions.

Findings

Established necessary and sufficient viability conditions for locally monotone evolution inclusions.

Proved well-posedness of lower semicontinuous solutions for Hamilton-Jacobi-Bellman equations.

Extended the theory to include a wider class of PDEs and control problems with state constraints.

Abstract

We establish necessary and sufficient conditions for viability of evolution inclusions with locally monotone operators in the sense of Liu and R\"ockner [J. Funct. Anal., 259 (2010), pp. 2902-2922]. This allows us to prove wellposedness of lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations associated to the optimal control of evolution inclusions. Thereby, we generalize results in Bayraktar and Keller [J. Funct. Anal., 275 (2018), pp. 2096-2161] on Hamilton-Jacobi equations in infinite dimensions with monotone operators in several ways. First, we permit locally monotone operators. This extends the applicability of our theory to a wider class of equations such as Burgers' equations, reaction-diffusion equations, and 2D Navier-Stokes equations. Second, our results apply to optimal control problems with state constraints. Third, we have uniqueness of viscosity solutions. Our results on viability and lower semicontinuous solutions are new even in the case of monotone operators.