The problems of state feedback and output feedback for fixed‐time stabilisation of a linear parabolic distributed parameter system with space‐dependent reactivity are considered by means of continuous backstepping approach and Lyapunov method. First, the invertible Volterra integral transformation with time dependent gain kernel is adopted to convert the original system into a fixed‐time stable target system with time‐dependent coefficient. The well‐posedness of the resulting kernel equations is proved by the method of successive approximation. Then a state feedback controller is designed to guarantee fixed‐time stabilisation of the closed‐loop system. Moreover, a fixed‐time observer is considered to estimate the state of the original system on the basis of the measurement signal at the boundary. Based on this observer, a observer‐based output feedback controller is established to fixed‐time stabilise the closed‐loop system in a prescribed time by using separation principle. Finally, a numerical simulation is provided to verify the feasibility of the proposed theoretical results by using the modified Ablowitz‐Kruskal‐Ladik scheme.

In this article, we are concerned with the exponential stabilization of parabolic distributed parameter systems with Neumann/mixed boundary conditions and external disturbance. To deal with the bounded disturbance, a sliding mode boundary control scheme is adopted based on the backstepping method. Firstly, the exponential stability of the considered system is proven along the selected sliding surface. Secondly, a sliding mode boundary controller is designed to regulate the closed‐loop system trajectory to a suitable sliding surface within a prescribed time and then maintain the sliding motion on the surface. Thirdly, on the chosen sliding surface, the closed‐loop system with the sliding mode boundary controller is exponentially stable. Finally, some numerical simulations are provided to verify the effectiveness of the theoretical results.

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