Stabilization of an ODE–Schrödinger Cascade

We consider the problem of stabilization of a linear ODE with input dynamics governed by the linearized Schrِdinger equation. The interconnection between the ODE and Schrِdinger equation is bi-directional at a single point. We construct an explicit feedback law that compensates the Schrِdinger dynamics at the inputs of the ODE and stabilizes the overall system. Our design is based on a two-step backstepping transformation by introducing an intermediate system and an intermediate control. By adopting the Riesz basis approach, the exponential stability of the closed-loop system is built with the pre-designed decay rate and the spectrum-determined growth condition is obtained. A numerical simulation is provided to illustrate the effectiveness of the proposed design.