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This paper deals with the null-controllability of a system of mixedparabolic-elliptic pdes at any given time $T>0$. More precisely, we considerthe Kuramoto-Sivashinsky--Korteweg-de Vries equation coupled with a secondorder elliptic equation posed in the interval $(0,1)$. We first show that thelinearized system is globally null-controllable by means of a localizedinterior control acting on either the KS-KdV or the elliptic equation. Usingthe Carleman approach we provide the existence of a control with the explicitcost $Ke^{K/T}$ with some constant $K>0$ independent in $T$. Then, applying thesource term method and the Banach fixed point argument, we conclude thesmall-time local null-controllability result of the nonlinear systems.  Besides, we also established a uniform null-controllability result for anasymptotic two-parabolic system (fourth and second order) that converges to theconcerned parabolic-elliptic model when the control is acting on the secondorder pde.

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