{"ID":2888079,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.00812","arxiv_id":"2508.00812","title":"On the controllability of the Kuramoto-Sivashinsky equation on multi-dimensional cylindrical domains","abstract":"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 $Ω_x=(0,a),$ $a\u003e0$ 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 $T_0(x_0)\u003e0$ such that the system is null controllable for all time $T \u003e T_0(x_0)$ 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 $T\u003cT_0(x_0).$ If we assume $x_0/a$ is an algebraic real number of order $d \u003e 1$, then we prove the controllability for any time $T\u003e0.$ 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.","short_abstract":"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 $Ω_x=(0,a),$ $a\u003e0$ 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 $\\{0\\}\\times ω$,...","url_abs":"https://arxiv.org/abs/2508.00812","url_pdf":"https://arxiv.org/pdf/2508.00812v2","authors":"[\"Víctor Hernández-Santamaría\",\"Subrata Majumdar\"]","published":"2025-08-01T17:45:59Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.AP\"]","methods":"[]","has_code":false}