Exponential stabilization of cascade ODE-linearized KdV system by boundary Dirichlet actuation

In this paper, we solve the problem of exponential stabilization for a class of cascade ODE-PDE systems governed by a linear ordinary differential equation and the 1−d linearized Korteweg–de Vries equation (KdV) posed on a bounded interval. The control for the whole system acts in the left boundary with Dirichlet condition of the KdV equation whereas the KdV acts in the linear ODE by a Dirichlet connection. We use the so-called backstepping method in infinite dimension to convert system under consideration to an exponentially stable cascade ODE-PDE system. Then, we use the invertibility of such design to achieve the exponential stability for the original ODE-PDE cascade system by using Lyapunov analysis.