Stabilization of an ODE-Schrödinger cascade

In this paper, we consider a problem of stabilization of an ODE-Schrodinger cascade, where the interconnection between them is bi-directional at a single point. By using the backstepping approach, which uses an invertible Volterra integral transformation together with the boundary feedback to convert the unstable plant into a well-damped target system, the target system in our case is given as a PDE-ODE cascade with exponential stability at the pre-designed decay rate. Instead of one-step backstepping control, which results in difficulty in finding the kernels, we develop a two-step backstepping control design by introducing an intermediate target system and an intermediate control. The exponential stability of the closed-loop system is investigated using the Lyapunov method. A numerical simulation is provided to illustrate the effectiveness of the proposed design.