On the partial stability problem for nonlinear discrete-time stochastic systems

We consider a system of nonlinear discrete-time equations subject to the influence of a discrete random process of the “white” noise type. It is assumed that the system admits a “partial” (with respect to some part of state variables) zero equilibrium. The problem of partial stability in probability is posed—the stability of a given equilibrium is not with respect to all but only to part of the variables determining it. To solve the problem, a discrete-stochastic version of the Lyapunov function method is used with the appropriate refinement of the requirements for the Lyapunov function. To expand the capabilities of the method used, it is proposed to correct the domain in which the auxiliary Lyapunov function is constructed; this is achieved by introducing an additional (vector, generally speaking) auxiliary function. Conditions of partial and asymptotic stability in probability in the indicated form are obtained. An example is given showing the specific features of the proposed approach.