Feedback stabilization of Convective Brinkman-Forchheimer Extended Darcy equations

Abstract

In this article, the following controlled convective Brinkman-Forchheimer extended Darcy (CBFeD) system is considered in a $d$-dimensional torus: \begin{align*} \frac{\partial\boldsymbol{y}}{\partial t}-\mu \Delta\boldsymbol{y}+(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}+\alpha\boldsymbol{y}+\beta\vert \boldsymbol{y}\vert ^{r-1}\boldsymbol{y}+\gamma\vert \boldsymbol{y}\vert ^{q-1}\boldsymbol{y}+\nabla p=\boldsymbol{g}+\boldsymbol{u},\ \nabla\cdot\boldsymbol{y}=0, \end{align*} where $d\in\{2,3\}$, $\mu,\alpha,\beta>0$, $\gamma\in\mathbb{R}$, $r,q\in[1,\infty)$ with $r>q\geq 1$. 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.