Boundary controllability for the Korteweg-de Vries-Burgers equation on a finite domain

In this paper, we consider the controllability of the Korteweg-de Vries-Burgers equation on a finite domain. Because the third-order of the Korteweg-de Vries-Burgers equation, there needs three boundary conditions to secure the well-posedness of the system. We show that if we only use the right Neumann boundary condition, then the linear system is exactly contrallable if and only if the length of the spatial domain does not belong to a set of critical values. By contrast, if we also use the right Dirichlet boundary condition besides the right Neumann boundary condition, the linear system is exactly controllable without the limit of the length of the domain. Moreover, we also show that the nonlinear system is local exactly controllable via the method of contraction mapping principle.