  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.
