Effective transmission conditions for Hamilton-Jacobi equations defined on two domains separated by an oscillatory interface

TL;DR

This paper analyzes the asymptotic behavior of optimal control problems with oscillatory interfaces, deriving effective transmission conditions for Hamilton-Jacobi equations as the oscillations vanish.

Contribution

It introduces a novel method to derive effective transmission conditions for Hamilton-Jacobi equations across oscillatory interfaces in the limit of vanishing oscillations.

Findings

Value functions converge to solutions with effective transmission conditions.

Effective conditions account for oscillations of the interface.

Results applicable to control problems with discontinuous dynamics.

Abstract

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across an oscillatory interface $\Gamma_\epsilon$. The oscillations of the interface have small period and amplitude, both of the order of $\epsilon$, and the interfaces $\Gamma_\epsilon$ tend to a straight line $\Gamma$. We study the asymptotic behavior as $\epsilon\to 0$. We prove that the value function tends to the solution of Hamilton-Jacobi equations in the two half-planes limited by $\Gamma$, with an effective transmission condition on $\Gamma$ keeping track of the oscillations of $\Gamma_\epsilon$.