Practical Fixed-Time Control of Switched Neutral Filippov Systems on Networks

In this article, the practical fixed-time (FxT) control of switched neutral Filippov systems (SNFSs) on networks is considered. The perturbation functions that can be discontinuous, and the neutral logics expressed by the difference operators, are addressed by new approaches. Several novel Lyapunov inequalities with indefinite functions are proposed, where detailed estimations of the settling-time (ST) are obtained by discussing the different values of the exponents of the Lyapunov functions, which can include the existing results. Considering that the states generally cannot converge to the origin accurately under finite-time control in real applications, practical FxT stability lemmas with indefinite functions are established for the first time, in which the bounded condition imposed on the indefinite function is more practical than the previously unbounded ones. By designing the adaptive control strategies, the FxT and practical FxT synchronization control are investigated based on the Lyapunov-Krasovskii functionals (LKFs), which show the delay characteristic via the adaptive update law containing delay values. Notably, the theoretical deficiency arising from the Lyapunov function when studying the FxT stability of real systems with delays by using the FxT stability lemmas with indefinite function is solved in a successful way. Finally, the validity of the main results is verified by numerical simulations on an electrical device containing an LC transmission line.