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

This paper deals with the null-controllability of a system of {\em mixed parabolic-elliptic pdes} at any given time $T>0$. More precisely, we consider the $\mathit{\text{Kuramoto-Sivashinsky–Korteweg-de}}\mathit{\text{Vries}}\mathit{\text{equation}}$ coupled with a second order elliptic equation posed in the interval $(0,1)$. We first show that the linearized system is globally null-controllable by means of a localized interior control acting on either the KS-KdV or the elliptic equation. Using the $\mathit{\text{Carleman}}\mathit{\text{approach}}$, we provide the existence of a control with the explicit cost $C{e}^{C\mathrm{/}T}$ with some constant $C>0$ independent in $T$. Then, applying the source term method followed by the $\mathit{\text{Banach}}\mathit{\text{fixed}}\mathit{\text{point}}\mathit{\text{theorem}}$, we conclude the small-time local null-controllability result of the nonlinear systems.