On the Partial Stability in the Probability of Nonlinear Stochastic Functional-Differential Systems with Aftereffect (Delay)

A system of nonlinear functional-differential equations with aftereffect (delay) subjected to random processes of “white” noise is considered. It is assumed that the system admits a partial (with respect to some part of the state variables) zero equilibrium position. The problem of the stability in the probability of the given equilibrium position is posed, and stability is considered not for all variables but for some of the variables that determine this equilibrium position. For the solution of this problem, a stochastic version of the method of Lyapunov–Krasovskii functionals is used with the appropriated specification of the requirements for the functionals. In order to expand the capabilities of the method used, it is also proposed to correct the domain of the functional space in which auxiliary Lyapunov–Krasovskii functionals are constructed. The conditions for partial stability of this type are obtained. Examples are given that show the features of the proposed approach.