On optimal upper bound for the settling time of fixed-time stable systems and its application in secure communication

This paper proposes a Lyapunov-based fixed-time stability (FXTS) theorem for nonlinear dynamical systems. An upper bound estimate of the settling time and the sufficient condition for FXTS of nonlinear systems are provided. An optimality criterion is derived to obtain the minimum value of the settling time’s upper bound estimate. Meanwhile, an insight into accurate estimation of the upper bound of settling time is provided. Based on the FXTS theorem, non-singular terminal sliding surfaces (NSTSM) and corresponding controllers are constructed to make the trajectories of the perturbed nonlinear dynamical system reach the equilibrium point robustly in some fixed-time. A secure communication scheme based on fixed-time synchronization of chaotic systems is also proposed as an application of the theoretical results obtained in this study. Numerical simulations are carried out for the illustration and validation of theoretical claims.