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Indefinite multiple Lyapunov functions of pth moment input‐to‐state stability and pth moment integral input‐to‐state stability for the nonlinear time‐varying stochastic systems with Markovian switching

The pth moment input‐to‐state stability and the pth moment integral input‐to‐state stability of stochastic systems with Markovian switching are investigated in this article. Sufficient conditions are provided by indefinite multiple Lyapunov functions (iMLFs) with a time‐varying sojourn time condition. These criteria quantify the stability of each individual subsystem with time‐varying infinitesimal generator of Lyapunov function, which allows infinitesimal generators to be positive in some intervals. An integral form of the sojourn time is introduced to replace the switch count process, which proposes a relaxed switching constraint. Uniformly asymptotic stability and uniformly exponential stability conditions with iMLFs are also presented. Some comparisons are given through theoretical derivations to show the advantages of the proposed results. Finally, two numerical examples verify the effectiveness and the advantages of the presented results.