Fixed‐time stabilisation of boundary controlled linear parabolic distributed parameter systems with space‐dependent reactivity

Abstract

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.

Results on Boundary Control for Parabolic Systems Using Backstepping Method

“…e state feedback and output feedback for fixed-time stabilization of a linear parabolic distributed parameter system with spacedependent reactivity is studied in [35]. A generalization of the scalar gradient extremum seeking algorithm, which maximizes static maps in the presence of parabolic PDEs, is presented in [36].…”