Local Null controllability of the Stabilized Kuramoto-Sivashinsky system using moment method

Abstract

This paper deals with the local null controllability of a coupled nonlinear parabolic system which is the Kuramoto-Sivashinsky-Korteweg-de Vries equation coupled with the heat equation through first order derivatives.More precisely, we prove the local null controllability of the system with a single localized interior control acting on either of the equations of the coupled system.Additionally, we also investigate the same issue with a single periodic boundary control acting through zeroth order derivatives of either of the components.We employ the well-known moment method to study the controllability of the corresponding linearized system around the origin with a suitable control cost$Ce^{\frac{C}{T}}$, for some$C > 0$ as$T\to 0^+ $.Further, the local null controllability of the original nonlinear system is shown by combining the source term method introduced in[50] and a Banach fixed point argument.