Finite-Time Stability of Switched Nonlinear Systems With Partially Unstable Modes via Hybrid Switching

This article investigates the finite-time stability (FTS) problem of switched nonlinear systems (SNSs) involving partially unstable modes under hybrid switching signals. In spite of significant challenge caused by unstable modes in SNSs, the qualitative analysis of FTS can still be conducted through the comprehensive utilization of switching information. To be specific, by constructing the discontinuous multiple Lyapunov function (DMLF) coupled with persistent dwell time (PDT)-based hybrid switching, a novel feasible solution for the FTS of SNSs is established. The “discontinuous” property of DMLF implies that certain state attenuation can be artificially designed at the switching instant to compensate for system divergence, and the proposed PDT-based switching employs respectively the τ -portion and theT-portion between stable and unstable subsystems to achieve FTS. Then, the FTS criteria are proposed in terms of stabilizing and destabilizing states during theT-portion, and the hybrid switching design scheme is established for stabilizing or destabilizing state during theT-portion. In particular, the overall settling-time upper bounds depending on both the initial state and the specific dwell time are further derived. Finally, two numerical examples including an application to the finite-time synchronization of switched complex dynamical networks are presented to demonstrate the practicality and effectiveness of the proposed theoretical results.