Control of a Fractional Distributed Parameter System with Non-constant Diffusivity

We consider the boundary feedback control of a fractional distributed parameter system (original system) with mixed boundary conditions and non-constant diffusivity. This situation can be viewed as a generalization of the boundary feedback stabilization problem of a fractional distributed parameter system with constant diffusivity. Specifically, we convert the original system into a general fractional distributed parameter system (new system) by a change of variables. We utilize the backsteping method and integral transformation to design a Dirichlet boundary feedback controller for the new system. Then, we can obtain a boundary feedback controller for the original system via the change of variables. Moreover, based on the Mittag-Leffler stability theory (fractional Lyapunov stability theory), we obtain sufficient conditions for Mittag-Leffler stability of the original system by the boundary feedback controller. Lastly, we present a specific numerical simulation example to verify the effectiveness of our proposed method.