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

Semi-Global and Global Fixed-Time Stability for Nonlinear Impulsive Systems

This study investigates the semi-global fixed-time stability (SGFTS) and global fixed-time stability (GFTS) of nonlinear impulsive systems (NISs). A key challenge in analyzing the SGFTS of such systems lies in the evolving integration methods caused by the impulses. To address this, we dynamically partition the semi-global attraction set (SGAS) and solve the corresponding differential equations within each subset. Additionally, by constructing the transition dynamics of impulse points and iteratively computing these points, we establish the conditions for SGFTS under both stabilizing and destabilizing impulses. For GFTS, the primary difficulty arises from the distinct trajectories and dynamics of points located inside and outside the SGAS. To overcome this, we introduce the concept of the maximum-minimum impulse interval and derive a sufficient condition that ensures the system can enter the SGAS from a distance under a finite number of impulses. Furthermore, we develop a criterion for GFTS under varying impulse degrees and provide convergence time estimation based on the research on SGFTS of NIS. Finally, numerical examples are presented to validate the theoretical results. Notably, in Example 3, a fixed-time impulse controller is designed based on the proposed theoretical framework to achieve global stabilization of complex systems. This example highlights the potential applications of this work in the field of control.

Fixed-time synchronization for two-dimensional coupled reaction-diffusion complex networks: Boundary conditions analysis

This paper examines fixed-time synchronization (FxTS) for two-dimensional coupled reaction-diffusion complex networks (CRDCNs) with impulses and delay. Utilizing the Lyapunov method, a FxTS criterion is established for impulsive delayed CRDCNs. Herein, impulses encompass both synchronizing and desynchronizing variants. Subsequently, by employing a Lyapunov-Krasovskii functional, two FxTS boundary controllers are formulated for CRDCNs with Neumann and mixed boundary condition, respectively. It is observed that vanishing Dirichlet boundary contributes to the synchronization of the CRDCNs. Furthermore, this study calculates the optimal constant for the Poincaré inequality in the square domain, which is instrumental in analyzing FxTS conditions for boundary controllers. Conclusive numerical examples underscore the efficacy of the proposed theoretical findings.