Moment Stability Conditions Research Articles

Necessary and sufficient conditions for moment stability of positive Markov jump linear systems

This paper focuses on the problems of stability for positive Markov jump linear systems (PMJLSs) and its application to nonlinear Markov jump stochastic systems (MJSSs). First, we provide sufficient and necessary conditions for p-moment exponential stability of PMJLS. Then, based on the proposed results on p-moment exponential stability of PMJLS, we obtain explicit stability criteria for nonlinear MJSSs. Finally, two numerical examples are presented to confirm the validity of the obtained theoretical results.

Stability of a Beddington–DeAngelis type predator–prey model with trichotomous noises

The stability analysis of a Beddington–DeAngelis (B–D) type predator–prey model driven by symmetric trichotomous noises is presented in this paper. Using the Shapiro–Loginov formula, the first-order and second-order solution moments of the system are obtained. The moment stability conditions of the B–D predator–prey model are given by using Routh–Hurwitz criterion. It is found that the stabilities of the first-order and second-order solution moments depend on the noise intensities and correlation time of noise. The first-order and second-order moments are stable when the correlation time of noise is increased. That is, the trichotomous noise plays a constructive role in stabilizing the solution moment with regard to Gaussian white noise. Finally, some numerical results are performed to support the theoretical analyses.

Asymptotic stability of structural systems based on Lyapunov exponents and moment Lyapunov exponents

An asymptotic expansion for the maximal Lyapunov exponent, the exponential growth rate of solutions to a linear stochastic system, and the moment Lyapunov exponent, the asymptotic growth rate of the moments of the response, have been obtained for systems driven by a small intensity real noise process. The systems under consideration are general four-dimensional dynamical systems with two critical modes. Almost-sure and moment stability conditions are obtained provided the natural frequencies of these critical modes are non-commensurable and the infinitesimal generator associated with the noise process has an isolated simple zero eigenvalue. In this paper, the results obtained are applied to a thin rectangular beam under the action of a stochastic follower force and a model of a vehicle traveling over a rough road. The stability regions predicted by the two different criteria are then compared.

Mean square stability of a rotating shaft under combined harmonic and stochastic excitations

The dynamic stability of a rotating shaft under parametric excitation consisting of a combination of harmonic terms and stationary stochastic processes is considered. The intensities of both harmonic and stochastic excitations and correlation time of stochastic excitations are assumed to be small in order to obtain approximate analytical results. Explicit stability conditions are derived for the first and second moments of a two-degree-of-freedom rotating shaft. When the stochastic excitation is a white noise excitation, the first moment stability conditions reduce to that of the deterministic case. It is shown that addition of non-white noise excitation has a stabilizing effect on the parametric instability of harmonically excited rotating shafts. Finally, the stability conditions of a symmetric shaft along with their numerical results are presented.

Almost sure and moments stability of jump linear systems

The stability of piecewise deterministic linear systems driven by an underlying finite Markov chain is analyzed. Necessary and sufficient conditions for moment stability are obtained by means of an explicit formula for the corresponding Liapunov exponent. The relationship between almost sure and moment stability is elucidated, revealing the possible occurrence of high order moment instability and large deviations from a stable sample path behavior.