On the controllability of the Kuramoto-Sivashinsky equation on multi-dimensional cylindrical domains

In this article, we investigate null controllability of the Kuramoto-Sivashinsky (KS) equation on a cylindrical domain $Ω=Ω_x\times Ω_y$ in $\mathbb R^N$, where $Ω_x=(0,a),$ $a>0$ and $Ω_y$ is a smooth domain in $\mathbb R^{N-1}$. We first study the controllability of this system by a control acting on $\{0\}\times ω$, $ω\subset Ω_y$, through the boundary term associated with the Laplacian component. The null controllability of the linearized system is proved using a combination of two techniques: the method of moments and Lebeau-Robbiano strategy. We provide a necessary and sufficient condition for the null controllability of this system along with an explicit control cost estimate. Furthermore, we show that there exists minimal time $T_0(x_0)>0$ such that the system is null controllable for all time $T > T_0(x_0)$ by means of an interior control exerted on $γ= \{x_0\} \times ω\subset Ω$, where $x_0/a\in (0,1)\setminus \mathbb{Q}$ and it is not controllable if $T<T_0(x_0).$ If we assume $x_0/a$ is an algebraic real number of order $d > 1$, then we prove the controllability for any time $T>0.$ Finally, for the case of $N=2 \text{ or } 3$, we show the local null controllability of the main nonlinear system by employing the source term method followed by the Banach fixed point theorem.