On stability of motion with respect in part of variables for stochastic systems

For stochastic control systems which are described by Ito's differential equations the problem of stability of motion with respect to part of variables is considered. In the case of linear stationary systems an algorithm for design of an auxiliary linear stationary system is described. The conditions for stability of the system with respect to all the variables are sufficient stability conditions with respect to part of variables of the initial systems. Classes of nonlinear systems are identified for which stability or otherwise of motion with respect to part of the variables can be established through analysis of the linear approximation. The results lead to a rule for design of a control signal in stabilization of motion with respect to part of the variables. Examples are given.