To Problem of Partial Stability of Nonlinear Discrete-Time Systems

Abstract

It is can identify three main classes of problems broadly characterizing partial stability of a dynamical systems, viz., (1) stability with respect to a part of the variables of the zero equilibrium position (Lyapunov-Rumyantsev partial stability problem), (2) stability of the "partial"zero equilibrium position, and (3) stability with respect to a part of the variables of the "partial"zero equilibrium position. In the problem of stability with respect to a part of the variables of the zero equilibrium position of systems of ordinary differential equations with continuous right-side assumes the domain of initial perturbations to be a sufficiently small neighborhood of the zero equilibrium position. Along with this statement, the case then initial perturbations can be large with respect to one part of non-controlled variables and arbitrary with respect to their other part is also considered. On the other hand, for stability problem of "partial" zero equilibrium positions of systems of ordinary differential equations also naturally assume that initial perturbations of variables that do not define the given equilibrium position can be large with respect to one part of the variables and arbitrary with respect to their other part. Contrary the assumptions that initial perturbations of this variables are either only arbitrary or only large the combined assumption made it possible an admissible trade-off between the meaning sense for notion of stability and the respective requirements on the Lyapunov functions. The article studies the problem of partial stability for nonlinear discrete-time systems: stability with respect to a part of the variables of "partial" 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 discrete 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 discrete-time systems is also discussed