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In this paper, we consider boundary feedback stabilization for unstable time fractional reaction diffusion equations. New state feedback controls with actuation on one end are designed by the backstepping method for both Dirichlet and Neumann boundary controls. By the Riesz basis approach and the fractional Lyapunov method, we prove the existence and uniqueness and the Mittag-Leffler stability for the closed-loop systems. For both cases, the observers and the observerbased output feedback are designed to stabilize the systems.

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In this paper, we consider boundary stabilization for a multi-dimensional wave equation with boundary control matched disturbance that depends on both time and spatial variables. The active disturbance rejection control (ADRC) approach is adopted in investigation. An extended state observer is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multidimensional system by infinitely many test functions. The disturbance is canceled in the feedback loop together with a collocated stabilizing controller. All subsystems in the closed-loop are shown to be asymptotically stable. In particular, the time varying high gain is first time applied to a system described by the partial differential equation for complete disturbance rejection purpose and the peaking value reduction caused by the constant high gain in literature. The overall picture of the ADRC in dealing with the disturbance for multi-dimensional partial differential equation is presented through this system. The numerical experiments are carried out to illustrate the convergence and effect of peaking value reduction.

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In this paper, the asymptotical stability for several classes of fractional order differential systems with time delay is investigated. We firstly present an integral inequality by which the Halanay inequality is extended to fractional order case. Based on the generalized Halanay inequality, we establish several asymptotical stability conditions under which the fractional order systems with time delay are asymptotically stable. It is worth to note that these stability conditions are easy to check without resorting to the solution expression of the systems.

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