Stochastic exponential and asymptotic stability of simple nonlinear systems (Q1390075)

The stochastic exponential \(p\)-stability and some types of stability in probability of dynamic systems, which are described by homogeneous systems of first-order differential equations with parametric perturbations, are analysed as a logical extension of exponential, asymptotic and weak stability of deterministic systems. The analysis of the stochastic stability is based on the second Lyapunov method constructing a suitable function, which is something like an upper estimate of the overall behaviour of the system. A Lyapunov function possessing certain properties, corresponding to the type of stochastic stability in question, is constructed. The author deduces the structure of other necessary and sufficient properties which the Lyapunov function has to satisfy, if the system is to be stable. Differences between analogous definitions in deterministic and stochastic domains are shown. The case of a nonlinear system is compared with linear and linearised systems, in order to decide, whether, and under which conditions the system can be linearised from the point of view of the analysis of stability, and whether the analysis can be performed using, for example, Routh-Hurwitz determinants. Some comments about a physical interpretation of the Lyapunov function are added, and about possibilities of its construction in individual cases.