On Stability with Respect to a Part of the Variables for Nonlinear Discrete-Time Systems with a Random Disturbances

Abstract

Nonlinear discrete (finite-difference) system of equations subject to the influence of a random disturbances of the"white" noise type, which is a difference analog of systems of stochastic differential equations in the Ito form, is considered.The increased interest in such systems is associated with the use of digital control systems, financial mathematics, as well aswith the numerical solution of systems of stochastic differential equations. Stability problems are among the main problemsof qualitative analysis and synthesis of the systems under consideration. In this case, we mainly study the general problemof stability of the zero equilibrium position, within the framework of which stability is analyzed with respect to all variablesthat determine the state of the system. To solve it, a discrete-stochastic version of the method of Lyapunov functions hasbeen developed. The central point here is the introduction by N. N. Krasovskii, the concept of the averaged finite differenceof a Lyapunov function, for the calculation of which it is sufficient to know only the right-hand sides of the system and theprobabilistic characteristics of a random process. In this paper, for the class of systems under consideration, a statement ofa more general problem of stability of the zero equilibrium position is given: not for all, but for a given part of the variables defining it. For the case of deterministic systems of ordinary differential equations, the formulation of this problem goes backto the classical works of A. M. Lyapunov and V. V. Rumyantsev. To solve the problem posed, a discrete-stochastic version ofthe method of Lyapunov functions is used with a corresponding specification of the requirements for Lyapunov functions. Inorder to expand the capabilities of the method used, along with the main Lyapunov function, an additional (vector, generally speaking) auxiliary function is considered for correcting the region in which the main Lyapunov function is constructed.