Adaptive stabilization to a class of coupled PDE-ODE systems

This paper is concerned with the adaptive stabilization for a class of uncertain coupled PDE-ODE systems. Firstly, a reversible infinite-dimensional backstepping transformation with appropriate kernel functions is introduced to change the original system into a new one, from which the control design becomes much convenient. Then, by Lyapunov method and some adaptive techniques, an adaptive stabilizing controller is successfully constructed, which guarantees that all the closed-loop system states are bounded while the original system states converging to zero. It is necessary to point out that, different from the closely literature, the ODE sub-system acts on the PDE sub-system at one end of the PDE domain rather than inside the domain. This makes the control design, particularly the derivation of the desirable kernel functions, much difficult.