Existence of viscosity solutions for Hamilton-Jacobi equations via Lyapunov control

TL;DR

This paper introduces a new approach to proving the existence of viscosity solutions for Hamilton-Jacobi equations using Lyapunov functions, comparison principles, and bounds on Hamiltonians, applicable to complex geometries and discontinuous dynamics.

Contribution

It develops a novel framework that relaxes regularity assumptions and incorporates Lyapunov control to establish viscosity solutions for Hamilton-Jacobi equations on manifolds and with discontinuous vector fields.

Findings

Established existence of viscosity solutions on Riemannian manifolds with boundary.

Extended the framework to Hamilton-Jacobi equations with discontinuous vector fields.

Provided a method that avoids regularity assumptions on solutions.

Abstract

We give a new perspective on the existence of viscosity solutions for a stationary and a time-dependent first-order Hamilton-Jacobi equation. Following recent comparison principles, we work in a framework in which we consider a subsolution and a supersolution for two equations in terms of two Hamiltonians that can be seen as an upper semi-continuous upper and lower semi-continuous lower bound of our original Hamiltonian respectively. The bounds are made rigorous in terms of Youngs inequality. The bounds are furthermore formulated in a way that incorporate a Lyapunov function which allows us to restrict part of the analysis to compact sets and to work with almost optimizers of the considered control problems. For this reason, we can relax assumptions on the control problem: most notably, we do not need completeness of set of controlled paths. Moreover, this strategy avoids a-priori analysis on the regularity of the candidate solutions. To complete our picture, we exhibit our result in two contexts. First, we consider Riemannian manifolds with smooth boundary, in which the dynamics allows for both "inward" and "outward" drift. The boundary conditions are embedded into the Hamiltonians itself. Second, we consider the solutions to a Fillipov differential equation, i.e. one with discontinuous vector field. We show that our notion of Hamiltonians leverages the natural embedding of the discontinuity in an associated set-valued differential inclusion.