The existence and stability of viscosity solutions to perturbed contact Hamilton-Jacobi equations

TL;DR

This paper proves the existence and stability of viscosity solutions for perturbed contact Hamilton-Jacobi equations, showing solutions converge to the original as perturbations vanish, with conditions for uniqueness and stability.

Contribution

It establishes the existence, convergence, and stability of viscosity solutions under perturbations for contact Hamilton-Jacobi equations, extending previous results to perturbed cases.

Findings

Viscosity solutions to perturbed equations exist and converge to the original solution.

Under certain conditions, the perturbed equation has a unique viscosity solution near the original.

The stability of solutions is preserved under perturbations, maintaining Lyapunov asymptotic stability.

Abstract

We consider a contact Hamiltonian $H(x,p,u)$ with certain dependence on the contact variable $u$. If $u_{-}$ is a viscosity solution of the contact Hamilton-Jacobi equation \[H(x,D_{x}u(x),u(x))=0,\quad x\in M,\] and $u_{-}$ is locally Lyapunov asymptotically stable, we will prove that the perturbed equation \[H(x,D_{x}u(x),u(x))+\varepsilon P(x,D_{x}u(x),u(x))=0,\quad x\in M,\] does exist viscosity solution $u_{-}^{\varepsilon}$ which converges uniformly to $u_{-}$, as perturbation parameter $\varepsilon$ converges to 0. Moreover, we give a case that in a neighborhood of viscosity solution $u_-$, the perturbed equation has an unique viscosity solution $u_{-}^{\varepsilon}$. Furthermore, $u_{-}^{\varepsilon}$ keeps locally Lyapunov asymptotically stability.