Asymptotic issue for porous media systems with linear multiplicative gradient-type noise via state constrained arguments

TL;DR

This paper establishes conditions for controlling stochastic porous media systems with gradient noise to stay within constraints, transforming the problem into a deterministic framework and applying viability theory for stabilization.

Contribution

It introduces a novel approach using rescaling to analyze viability and stabilization of stochastic porous media systems with divergence type noise.

Findings

Derived necessary and sufficient conditions for system viability.

Developed a framework for stabilization of stochastic porous media equations.

Transformed stochastic systems into random deterministic ones for analysis.

Abstract

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 continuation of the results in [10], as we consider the case of divergence type noise perturbation. On the other hand, it provides a different framework in which the quasi-tangency condition can be obtained with optimal speed. In comparison with the aforementioned result, here we transform the stochastic system into a random deterministic one, via the rescaling approach, then we study the viability of random sets. As an application, conditions for the stabilization of the stochastic porous media equations are obtained.