On state-constrained porous-media systems with gradient-type multiplicative noise

The aim of the present paper is to provide necessary and sufficient conditions to maintain a stochastic coupled system with porous media components and gradient-type noise in a prescribed set of constraints by using internal controls. This work is a complementary contribution to the results obtained by the same authors, also on the viability problem associated to the porous media equation, but with Lipschitz noise. Second, the present paper provides a different framework in which the quasi-tangency condition can be obtained with optimal speed. In comparison with the aforementioned result, and from a technical point of view, here, we transform the stochastic system into a random-PDE one, via the rescaling approach, and then we study the viability of random sets. As an application, (stronger) conditions for the stabilization of the stochastic porous media equations are obtained. These are illustrated on a simple example.