Existence of viscosity solutions for Hamilton-Jacobi equations via Lyapunov control
Serena Della Corte, Richard C. Kraaij
TL;DR
This work develops a Lyapunov-controlled framework that uses upper and lower Hamiltonian envelopes $\mathcal{H}_\dagger$ and $\mathcal{H}_\ddagger$ to establish the existence of viscosity solutions for stationary and time-dependent Hamilton–Jacobi equations on manifolds, including cases with boundaries and discontinuous vector fields. By bounding the original Hamiltonian via Young’s inequality and enforcing compact containment with a Lyapunov function $\Upsilon$, the authors prove that the value functions yield viscosity sub- and supersolutions to augmented operators $H^\Upsilon_\dagger$ and $H^\Upsilon_\ddagger$ in both the stationary equation $u - \lambda H^\Upsilon_\dagger u = h$ and the evolutionary equation $\partial_t u + \kappa u - H^\Upsilon_\dagger u = 0$ (with the corresponding supersolutions using $H^\Upsilon_\ddagger$). The framework accommodates smooth manifolds with boundary and Filippov differential equations by embedding boundary behavior into the envelope Hamiltonians and leveraging a dynamic programming principle, growth bounds, and local regularity arguments that avoid requiring a priori regularity of the value functions. Key contributions include the construction of boundary-adapted Hamiltonians, a basic comparison principle in geodesically convex domains, and a robust existence theory for sub- and supersolutions under minimal regularity assumptions. This advances viscosity-solution existence theory to unbounded domains and non-smooth dynamics, with implications for controlled dynamics, boundary conditions, and discontinuous vector fields.
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.