Stabilization of hybrid PDE-ODE systems with harmonic disturbances

Adaptive boundary stabilization is investigated for a class of hybrid PDE-ODE systems with uncertain harmonic disturbances. The essential difference between this paper and the existing related literature is the presence of the unknown disturbances belonging to an unknown interval, which makes the problem unsolved so far. To solve the problem, by the Lyapunov method and adaptive technique, an adaptive controller is first constructed. Then, by the semigroup approach, the well-posedness of the closed-loop system is established. Finally, by LaSalle´s invariance principle, we show that the designed controller guarantees that the resulting closed-loop system is asymptotically stable, and furthermore the parameter estimates are shown to ultimately converge to their own real values.