Backstepping-based boundary control design for a fractional reaction diffusion system with a space-dependent diffusion coefficient. (September 2018) - Chen, Juan; Cui, Baotong; Chen, YangQuan
Abstract: This paper presents a boundary feedback control design for a fractional reaction diffusion (FRD) system with a space-dependent (non-constant) diffusion coefficient via the backstepping method. The contribution of this paper is to generalize the results of backstepping-based boundary feedback control for a FRD system with a space-independent (constant) diffusion coefficient to the case of space-dependent diffusivity. For the boundary stabilization problem of this case, a designed integral transformation treats it as a problem of solving a hyperbolic partial differential equation (PDE) of transformation's kernel, then the well posedness of the kernel PDE is solved for the plant with non-constant diffusivity. Furthermore, by the fractional Lyapunov stability (Mittag-Leffler stability) theory and the backstepping-based boundary feedback controller, the Mittag-Leffler stability of the closed-loop FRD system with non-constant diffusivity is proved. Finally, an extensive numerical example for this closed-loop FRD system with non-constant diffusivity is presented to verify the effectiveness of our proposed controller. Highlights: The backstepping method is concerned for the boundary control problem of a fractional reaction diffusion (FRD) system with space-dependent (non-constant) diffusivity. By an integral transformation, the boundary stabilization problem is converted into a problem of solving a hyperbolic PDE with transformation's kernel. The kernel k(x, y) with nonzero-value and zero-value k(0, 0) is analyzed and the well posedness of the kernel PDE is solved for the system with non-constant diffusivity. Based on the fractional Lyapunov (Mittag-Leffler) stability theory and the boundary feedback controller, the Mittag-Leffler stability of the closed-loop system is proved. An extensive numerical example for the FRD system with non-constant diffusivity is used to verify the effectiveness of our proposed controller.