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

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 $ϵ$ while the period is of the order of $ϵ$ 2. As $ϵ$ $\to$ 0, the interfaces tend to a straight line $\mathrm{\Gamma }$. We study the asymptotic behavior of the value function as $ϵ$ $\to$ 0. We prove that the value function tends to the solution of Hamilton-Jacobi equations in the two half-planes limited by $\mathrm{\Gamma }$, with an effective transmission condition on $\mathrm{\Gamma }$ keeping track of the oscillations.