Boundary stabilization and disturbance rejection for a time fractional order diffusion-wave equation

In this paper, we study the boundary stabilization and disturbance rejection for an unstable time fractional diffusion-wave equation involving Caputo time fractional derivative. When there is no boundary external disturbance, both state feedback control and output feedback control via boundary actuation are proposed by the classical backstepping method. It is proved that the state feedback makes the closed-loop system Mittag-Leffler stable while the output feedback makes the closed-loop system asymptotically stable. When there is boundary external disturbance, we propose a disturbance estimator which is constructed by two infinite dimensional auxiliary systems to recover the external disturbance. The resulting closed-loop system is Mittag-Leffler stable and the states of all subsystem involved are uniformly bounded. As a byproduct, we solve rigorously completely the two longtime unsolved problems raised in [Nonlinear Dynam., 38(2004), 339-354] where all the results are only verified by simulations.