Homogenization of a transmission problem with Hamilton-Jacobi equations and a two-scale interface. Effective transmission conditions

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 homogenization framework for Hamilton-Jacobi equations with two-scale oscillatory interfaces, establishing effective transmission conditions in the limit.

Findings

Value function converges to a solution with effective transmission conditions.

Derived explicit form of the effective transmission condition.

Validated the homogenization approach for oscillatory interfaces.

Abstract

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across a two-scale oscillatory interface. Typically, the amplitude of the oscillations is of the order of $\epsilon$ while the period is of the order of $\epsilon$ 2. As $\epsilon$ $\rightarrow$ 0, the interfaces tend to a straight line $\Gamma$. We study the asymptotic behavior of the value function as $\epsilon$ $\rightarrow$ 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.