  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.
