Stochastic stability of positive Markov jump subhomogeneous and homogeneous nonlinear systems

In this paper, we investigate the stochastic stability of positive Markov jump nonlinear systems (PMJNS) with homogeneous or subhomogeneous properties, extending the existing results for positive Markov jump linear systems (PMJLS). By introducing sum-separable Lyapunov functions and an infinitesimal generator, we derive novel stability criteria that mitigate the impact of Markov jump-induced coupling terms through relationships among Lyapunov functions, weighted l ∞ norms, and max-norms. A key lemma is established to analyze solution bounds for subhomogeneous systems, enabling a three-case stability analysis based on the homogeneity degree τ (i.e., 0 < τ < 1 , τ = 1 , and τ > 1 ). Our results show that the stochastic stability of PMJNS can be determined solely by the algebraic properties of nonlinear functions and the probability characteristics of Markov chains, relaxing the strict component-wise negativity requirements in prior works. Specifically, for homogeneous systems (a special case of subhomogeneous systems), simplified stability criteria are derived. Two numerical examples (including a multi-area power system model) validate the theoretical findings and demonstrate the approach’s effectiveness.