Asymptotic stability of a class of nonlinear stochastic systems undergoing Markovian jumps

• The largest Lyapunov exponent of the Markovian-jump system is derived.
• The largest Lyapunov exponent is a convex linear combination of different forms.
• Necessary and sufficient conditions for asymptotic stability are obtained.
• Stability conditions prescribe limitations on occupancy time in each unstable form.

Systems which specifications change abruptly and statistically, referred to as Markovian-jump systems, are considered in this paper. An approximate method is presented to assess the asymptotic stability, with probability one, of nonlinear, multi-degree-of-freedom, Markovian-jump quasi-nonintegrable Hamiltonian systems subjected to stochastic excitations. Using stochastic averaging and linearization, an approximate formula for the largest Lyapunov exponent of the Hamiltonian equations is derived, from which necessary and sufficient conditions for asymptotic stability are obtained for different jump rules. In a Markovian-jump system with unstable operating forms, the stability conditions prescribe limitations on time spent in each unstable form so as to render the entire system asymptotically stable. The validity and utility of this approximate technique are demonstrated by a nonlinear two-degree-of-freedom oscillator that is stochastically driven and capable of Markovian jumps.