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

TL;DR

This paper studies the null controllability of the multi-dimensional Kuramoto-Sivashinsky equation on cylindrical domains, providing conditions, control costs, and minimal times for controllability, including nonlinear cases.

Contribution

It establishes necessary and sufficient conditions for null controllability of the KS equation on cylindrical domains, including explicit control costs and minimal times, and extends results to nonlinear systems.

Findings

Null controllability characterized by explicit conditions.

Existence of minimal control time depending on initial position.

Local null controllability for nonlinear systems in certain dimensions.

Abstract

In this article, we investigate null controllability of the Kuramoto-Sivashinsky (KS) equation on a cylindrical domain $\Omega=\Omega_x\times \Omega_y$ in $\mathbb R^N$, where $\Omega_x=(0,a),$ $a>0$ and $\Omega_y$ is a smooth domain in $\mathbb R^{N-1}$. We first study the controllability of this system by a control acting on $\{0\}\times \omega$, $\omega\subset \Omega_y$, 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 $\gamma = \{x_0\} \times \omega \subset \Omega$, 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.