  This paper deals with the null-controllability of a system of mixed
parabolic-elliptic pdes at any given time $T>0$. More precisely, we consider
the Kuramoto-Sivashinsky--Korteweg-de Vries 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 Carleman approach we provide the existence of a control with the explicit
cost $Ke^{K/T}$ with some constant $K>0$ independent in $T$. Then, applying the
source term method and the Banach fixed point argument, we conclude the
small-time local null-controllability result of the nonlinear systems.
  Besides, we also established a uniform null-controllability result for an
asymptotic two-parabolic system (fourth and second order) that converges to the
concerned parabolic-elliptic model when the control is acting on the second
order pde.
