Global boundary stabilization to trajectories of the deterministic and stochastic porous-media equation

Here we deal with the problem of boundary asymptotic exponential stabilization of flows through porous media. More exactly we study the porous media equation with general monotone porosity in a bounded domain of dimension $ d = 1,2,3 $. We construct an explicit, linear, of finite-dimensional structure feedback controller with Dirichlet part-boundary actuation, which stabilizes any trajectory of the system, for any given initial data. The form of the controller is based on the spectrum of the Dirichlet-Laplace operator and ensures exponential decay to zero of the fluctuation variable for any a priori prescribed decay rate. Also, we extend these results to the case of porous media equation perturbed by Itô Lipschitz noise.