Stabilization and Stability Robustness of Coupled Non-Constant Parameter Time Fractional PDEs

Abstract

This paper considers observer-based output feedback stabilization and stability robustness against small diffusivity perturbations of coupled time fractional partial differential equations (PDEs) with space-dependent (non-constant) parameters. Herein, the plant is equipped with the only available measurement at x = 0 and actuation at x = 1. By backstepping transformation, the well-posedness of the kernel matrix PDE and the observer gains are obtained. Then an output feedback controller is introduced and the Mittag-Leffler stability of the closed-loop system is proved by the fractional Lyapunov method. Robustness analysis of diffusion coefficients uncertainty (with a small perturbation in the diffusion coefficient) is also provided. The output feedback stabilization of the closed-loop system is tested by a numerical example. INDEX TERMS Coupled time fractional PDEs, output feedback boundary stabilization, uncertainty analysis, space-dependent coefficients.

Boundary output feedback stabilization for spacial multi‐dimensional coupled fractional reaction–diffusion systems

“…Remark Output feedback stabilization for fractional PDEs with one‐dimensional spacial variable was studied in several works [9‐13], but they did not investigate how to cope with the multidimensional spacial variable. Considering that the practical significance and physical description of extending the results for PDEs to coupled systems with multidimensional spacial variables are more important, Theorems 2 and 3 provide the stabilization design of spacial multidimensional fractional PDEs.…”

“…The Mittag-Leffler stabilization of a fractional heat equation with boundary external disturbance was discussed in [10]. More recently, an output feedback control was designed for fractional diffusion systems with spatially varying parameters in [11], while the stability robustness against small diffusivity perturbations was further studied in [12]. For more about the stabilization of fractional systems and the boundary control of PDEs, we refer to several works [13][14][15][16][17][18][19][20][21].…”

Observation and stabilisation of coupled time‐fractional reaction–advection–diffusion systems with spatially‐varying coefficients

“…by employing (28) and observing the structure of H(y). Moreover, R 10 is obtained by evaluating (17) with (20) (x = 0) and (22), namely…”

“…After applying the integration by parts, (54)-(56), the Cauchy-Schwarz inequality, Young's inequality to (59) and taking account into BCs (22) and 23, we have…”

“…This methodology has also been applied to fractional‐order systems with different boundary conditions (BCs) for boundary control problems, as done in [20, 21]. However, little research effort has been attracted to the observation and output feedback stabilisation problems of coupled time‐fractional PDE systems with spatially‐varying coefficients expect for the work in [13, 22]. Motivated by the significance of fractional derivative model in the above argument and the special choice of the target system for the stabilisation of integer‐order PDEs in the work [23], we concern the output feedback stabilisation problem for coupled time‐fractional RAD (FRAD) systems with space‐dependent coefficients by the backstepping approach.…”