Boundary stabilization and disturbance rejection for an unstable time fractional diffusion-wave equation

¹ School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha 410075, China.
² School of Mathematics and Big Data, Foshan University, Foshan 528000, China.
³ School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China.
⁴ Mechatronics, Embedded Systems and Automation Lab, University of California, Merced 95343, CA, USA.

^* Corresponding author: zehaowu@amss.ac.cn

Received: 15 February 2021
Accepted: 4 January 2022

Abstract

In this paper, we study boundary stabilization and disturbance rejection problem for an unstable time fractional diffusion-wave equation with Caputo time fractional derivative. For the case of no boundary external disturbance, both state feedback control and output feedback control via Neumann boundary actuation are proposed by the classical backstepping method. It is proved that the state feedback makes the closed-loop system Mittag-Leffler stable and the output feedback makes the closed-loop system asymptotically stable. When there is boundary external disturbance, we propose a disturbance estimator constructed by two infinite dimensional auxiliary systems to recover the external disturbance. A novel control law is then designed to compensate for the external disturbance in real time, and rigorous mathematical proofs are presented to show that the resulting closed-loop system is Mittag-Leffler stable and the states of all subsystems involved are uniformly bounded. As a result, we completely resolve, from a theoretical perspective, two long-standing unsolved mathematical control problems raised in Liang [Nonlinear Dyn. 38 (2004) 339–354] where all results were verified by simulations only.