  Cut meshes are a type of mesh that is formed by allowing embedded boundaries
to "cut" a simple underlying mesh resulting in a hybrid mesh of cut and
standard elements. While cut meshes can allow complex boundaries to be
represented well regardless of the mesh resolution, their arbitrarily shaped
and sized cut elements can present issues such as the small cell problem, where
small cut elements can result in a severely restricted CFL condition. State
redistribution, a technique developed by Berger and Giuliani [1], can be used
to address the small cell problem. In this work, we pair state redistribution
with a high-order discontinuous Galerkin scheme that is $L_2$ energy stable for
arbitrary quadrature. We prove that state redistribution can be added to a
provably $L_2$ energy stable discontinuous Galerkin method on a cut mesh
without damaging the scheme's $L_2$ stability. We numerically verify the high
order accuracy and stability of our scheme on two-dimensional wave propagation
problems.
