Fixed-Time Stability of Nonlinear Systems With Destabilizing Delayed Impulses: Necessary and Sufficient Conditions

This paper investigates the problem of fixed-time stability of nonlinear systems subject to destabilizing delayed impulses. The existing literature usually uses inequality-based techniques to deal with impulses and delays, which leads to considerable conservatism, especially when systems contain large impulsive gains or large delays. In contrast, inspired by the Möbius transformation, this article presents necessary and sufficient criteria for fixed-time stability of nonlinear systems subject to periodic destabilizing impulses with delay, or even with arbitrarily finite delays. To conquer the challenge caused by large delays, this article proposes a generic framework that equivalently transforms a system with large delayed impulses into a series of systems with small delayed impulses. The equivalence of this transformation does not induce any conservatism. The criteria reveal that as the delay increases, the system switches between stability and instability. In addition, it proved that the impulsive systems under some conditions exhibit a subcritical Hopf bifurcation: If the impulsive frequency is less than a critical value, then the system converges globally to an equilibrium point in a fixed time; otherwise the system oscillates periodically. Finally, examples of circuit systems are used to validate the theoretical results.