Fixed-time stability for nonlinear systems with large delayed impulses: delayed impulsive sequence decomposition

This paper addresses the fixed-time stability (FxTS) problem for nonlinear dynamical systems impacted by delayed impulses, specifically in the case where delay durations exceed the impulsive intervals. Traditional approaches often handle large delays through inequality-based techniques, resulting in considerable conservatism. To address such an issue, this paper introduces a new delayed impulsive sequence decomposition method, enabling a transformation that models systems with large delays as a set of equivalent systems with small delays, where delays are less than impulsive intervals. This reformulation reveals that the conservatism for large-delays criteria depends solely on small-delays criteria. Building on this insight, the paper develops less conservative FxTS criteria for systems with both uniform and general impulsive sequences. Additionally, this paper examines the necessary condition for FxTS in systems experiencing destabilizing impulses. A notable finding is the dual effect of impulsive delays: while a moderate increase in delay can stabilize an otherwise unstable system within a fixed time, excessive delay can destabilize a system that was initially fixed-time stable. Finally, the theoretical findings are verified through four numerical examples.