On Problem of Stability in Probability for "Partial" Equilibrium Positions of Nonlinear Stochastic Systems

Abstract

The partial stability problems naturally arise in applications either from the requirement of proper performance of a system or in assessing system capability. In addition, a lot of actual (or desired) phenomena can be formulated in terms of these problems and be analyzed with these problems taken as the basis. The following multiaspect phenomena and problems can be indicated: adaptive stabilization; spacecraft stabilization (especially stabilization by rotors); drift of the gyroscope axis; Lotka-Volterra ecological principle, e.t.c. Also very effective is the approach to the problem of stability with respect to all variables based on preliminary analysis of partial stability. The article studies the problem of partial stability for nonlinear stochastic systems of differential equations Ito: stability with respect to a part of the variables in probability of "partial" zero equilibrium position. Initial perturbations of variables that do not define the given equilibrium position can be large (belonging to an arbitrary compact set) with respect to one part of the variables and arbitrary with respect to their other part. A conditions of stability of this type are obtained in the context of a stochastic analog of the Lyapunov functions method, which generalize a number of existing results. Example is given. The problem of unification of process of studying partial stability problems of stationary and non-stationary nonlinear stochastic systems of differential equations is also discussed.