On the controllability of a system coupling Kuramoto–Sivashinsky–Korteweg–de Vries and transport equations

In this paper, we study the null controllability of a coupled parabolic–hyperbolic system in one dimension with a single control using the moment method. More precisely, we consider a system coupling Kuramoto–Sivashinsky–Korteweg–de Vries equation and transport equation through first-order derivatives. We explore the null controllability of four different control systems with the control acting either on the periodic boundary or in some open subset of the interior of the domain with periodic boundary conditions. Depending on the position of the control, we get some regular periodic Sobolev space as the space of initial data for which the null controllability holds, provided the time is sufficiently large.