Stochastic exponential and asymptotic stability of simple non-linear systems

Abstract

The stochastic exponential p-stability and some types of stability in probability of dynamic systems, which is described by a normal homogeneous system of differential equations of the first order with parametric perturbations, is analysed as a logical extension of the exponential, asymptotic and weak stability of deterministic systems. The analysis of the stochastic stability has been based on the second Lyapunov method. The Lyapunov function possessing certain properties, corresponding to the type of stochastic stability in question, is constructed. This function must always be positive definite, and the result of the application of the adjoined Fokker-Planck-Kolgomorov operator must be the function of negative or zero value, with the exception of the origin. The paper deduces the structure of other necessary and sufficient properties which the Lyapunov function has to satisfy, if the system is to be stable. The differences between analogous definitions in deterministic and stochastic domains have been shown. The case of a non-linear system is compared with the 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 the stability, and whether the analysis can be performed using, for example, the Rous-Hurwitz determinants. The fundamental problem is of course the level of stability which the simplified fictitious system has to satisfy, in order to imply the required type of stability of the original system. This alignment is, of course, limited on such a type of non-linearities, which do not cause energetically unstable branches, multiple equilibrium states and the polymodal character of the response. Some comments about the physical interpretation of the Lyapunov function have been added, as well as about the possibilities of its construction in individual cases.