Indirect stabilization of weakly coupled systems

We investigate stability properties of indirectly damped systems of evolution equations in Hilbert spaces, under new compatibility assumptions. We prove polynomial decay for the energy of solutions and optimize our results by interpolation techniques, obtaining a full range of power-like decay rates. In particular, we give explicit estimates with respect to the initial data. We discuss several applications to hyperbolic systems with hybrid boundary conditions, including the coupling of two wave equations subject to Dirichlet and Robin type boundary conditions, respectively. Furthermore, we show how these tecniques can be applied to the same problem when the coupling occurs on a subset of the boundary. In this case, either exponential or polynomial decay rate might be expected, depending on suitable compatibility conditions.