Stabilization of coupled PDE-ODE system with spatially varying coefficient

Stabilization is investigated for a class of coupled PDE-ODE systems with spatially varying coefficient. The systems under investigation are more general than those in the existing closely related literature since the coefficient of coupled term in the PDE sub-system is spatially varying. By infinite-dimensional backstepping method, the state-feedback stabilizing controller is explicitly constructed, which guarantees that the closed-loop system is exponentially stable in the sense of certain norm. It is worthwhile pointing out that, the method of successive approximation is adopted to derive the kernel functions of the backstepping transformation, since the presence of the spatially varying coefficient makes the kernel equation much more difficult to solve by the existing methods. A simulation example is presented to illustrate the effectiveness of the proposed method.