Boundary stabilisation of an unstable parabolic PDE with a time-varying domain and the external disturbance

This paper considers the stability of the one-dimensional parabolic system, where one end changes over time and the other is the control end with the external disturbance. Firstly, by the boundary immobilisation method, the displacement change of the system boundary is transferred into the equation so that the original system is transformed into a system with fixed boundaries. Secondly, by combining the backstepping transformation and the sliding mode control method, the feedback control is proposed to compensate the instability of the system itself and reject the matched disturbance. Then, the resulting closed-loop system will be in the form of (Formula presented.), where (Formula presented.) generates a (Formula presented.) semigroup, (Formula presented.) and (Formula presented.) are bounded and unbounded operators respectively, and (Formula presented.) is the external input. The existence of the generalised solution to the closed-loop system is proved by using the eigenfunction expansion of the system solution. By the Lyapunov method, the closed-loop system is shown to be exponentially stable. Finally, some numerical simulations are presented to illustrate the effectiveness of the proposed controller.