Feedback Stabilization of Convective Brinkman-Forchheimer Extended Darcy Equations

In this article, the following controlled convective Brinkman-Forchheimer extended Darcy (CBFeD) system is considered in a d-dimensional torus:

$$\begin{aligned} \frac{\partial {\varvec{y}}}{\partial t}-\mu \Delta {\varvec{y}}+({\varvec{y}}\cdot \nabla ){\varvec{y}}+\alpha {\varvec{y}}+\beta \vert {\varvec{y}}\vert ^{r-1}{\varvec{y}}+\gamma \vert {\varvec{y}}\vert ^{q-1}{\varvec{y}}+\nabla p={\varvec{g}}+{\varvec{u}},\ \nabla \cdot {\varvec{y}}=0, \end{aligned}$$

where \(d\in \{2,3\}\), \(\mu ,\alpha ,\beta >0\), \(\gamma \in {\mathbb {R}}\), \(r,q\in [1,\infty )\) with \(r>q\ge 1\). We prove the exponential stabilization of CBFeD system by finite- and infinite-dimensional feedback controllers. The solvability of the controlled problem is achieved by using the abstract theory of m-accretive operators and density arguments. As an application of the above solvability result, by using infinite-dimensional feedback controllers, we demonstrate exponential stability results such that the solution preserves an invariance condition for a given closed and convex set. By utilizing the unique continuation property of controllability for finite-dimensional systems, we construct a finite-dimensional feedback controller which exponentially stabilizes CBFeD system locally, where the control is localized in a smaller subdomain. Furthermore, we establish the local exponential stability of CBFeD system via proportional controllers.