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

This paper deals with the null-controllability of a system mixed parabolic-elliptic pdes at any given time T>0. More precisely, we consider Kuramoto-Sivashinsky–Korteweg-de Vries equation coupled second order elliptic posed in interval (0,1). We first show that linearized is globally null-controllable by means localized interior control acting on either KS-KdV or equation. Using Carleman approach, provide existence explicit cost CeC/T some constant C>0 independent T. Then, applying source term method developed [39], followed Banach fixed point theorem, conclude small-time local result nonlinear system.