Local energy decay for the wave equation with a time-periodic non-trapping metric and moving obstacle

Consider the mixed problem with Dirichelet condition associated to the wave equation $\partial_t^2u-\Div_{x}(a(t,x)\nabla_{x}u)=0$, where the scalar metric $a(t,x)$ is $T$-periodic in $t$ and uniformly equal to 1 outside a compact set in $x$, on a $T$-periodic domain. Let $\mathcal U(t, 0)$ be the associated propagator. Assuming that the perturbations are non-trapping, we prove the meromorphic continuation of the cut-off resolvent of the Floquet operator $\mathcal U(T, 0)$ and establish sufficient conditions for local energy decay.