Finite-time convergence of solutions of Hamilton-Jacobi equations

TL;DR

This paper proves that solutions to certain Hamilton-Jacobi equations can be reached from other solutions within finite time, under specific conditions on the Hamiltonian.

Contribution

It establishes finite-time convergence of viscosity solutions for Hamilton-Jacobi equations with decreasing Hamiltonians satisfying Tonelli conditions.

Findings

Viscosity solutions can be reached in finite time

Results apply to Hamiltonians decreasing in u

Provides conditions for finite-time solution convergence

Abstract

Suppose that $H(x,u,p)$ is strictly decreasing in $u$ and satisfies Tonelli conditions in $p$. We show that each viscosity solution of $H(x,u,u_x)=0$ can be reached by many viscosity solutions of $$ w_t+H(x,w,w_x)=0, $$ in a finite time.