Fixed-Time Stability of Nonlinear Impulsive Systems and Its Application to Inertial Neural Networks

This article is concerned with the fixed-time stability (FTS) problem of nonlinear impulsive systems (NISs). By means of the impulsive control mechanism and Lyapunov functions theory, several sufficient conditions are established to ensure the FTS of general NISs. Meanwhile, some novel impulse-dependent settling-time estimation schemes are developed, which fully considers the influence of stabilizing impulses and destabilizing impulses on the convergence rate of the system states. The proposed schemes establish a quantitative relationship between the upper bound of the settling time and impulse effects. It shows that stabilizing impulses can accelerate the convergence rate of the system states and leads to the upper bound of the settling time being smaller. Conversely, destabilizing impulses can reduce it and make the upper bound of the settling time larger. Then, the theoretical results are applied to delayed inertial neural networks (DINNs), where two kinds of controllers are designed to realize fixed-time synchronization of the considered systems in the impulse sense. Finally, some numerical examples are provided to illustrate the validity of the proposed theoretical results.

Automated summary

This article investigates the problem of fixed-time stability in nonlinear impulsive systems (NISs) using impulsive control and Lyapunov functions theory. The article establishes sufficient conditions for the fixed-time stability of general NISs and proposes novel impulse-dependent settling-time estimation schemes that consider the influence of stabilizing and destabilizing impulses on system convergence rate. The proposed schemes provide a quantitative relationship between the upper bound of settling time and impulse effects, showing that stabilizing impulses accelerate convergence while destabilizing impulses decrease it. The theoretical results are applied to delayed inertial neural networks (DINNs) to achieve fixed-time synchronization using two types of controllers. Numerical examples are provided to validate the proposed theoretical results.