Controllability of a simplified time-discrete stabilized Kuramoto-Sivashinsky system

Abstract

<p style='text-indent:20px;'>In this paper, we study some controllability and observability properties for a coupled system of time-discrete fourth- and second-order parabolic equations. This system can be regarded as a simplification of the well-known stabilized Kumamoto-Sivashinsky equation. Unlike the continuous case, we can prove only a relaxed observability inequality which yields a <inline-formula><tex-math id="M1">\begin{document}$ \phi(\triangle t) $\end{document}</tex-math></inline-formula>-controllability result. This result tells that we cannot reach exactly zero but rather a small target whose size goes to 0 as the discretization parameter <inline-formula><tex-math id="M2">\begin{document}$ \triangle t $\end{document}</tex-math></inline-formula> goes to 0. The proof relies on a known Carleman estimate for second-order time-discrete parabolic operators and a new Carleman estimate for the time-discrete fourth-order equation.</p>

Local null-controllability of a system coupling Kuramoto-Sivashinsky-KdV and elliptic equations

“…The author in [47] proved the boundary local nullcontrollability of the KS equation by utilizing the source term method (see [42]) followed by the Banach fixed point argument where a suitable control cost Ce C/T of the linearized model plays the crucial role. Similar strategy has been applied in [35] to study the boundary local null-controllability of a simplified stabilized KS system (see below about this system). In this regard, we must mention that the nonlinearity uu x in the KS system also appears in the Burgers equation: u t − u xx + uu x = 0, and concerning the controllability results of this equation we address the following works: [24], [30], [29].…”