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# Statistical characteristics of attainable set of controllable system, non-wandering, and minimal attraction center

2009, Vol. 5, No. 2, pp.  265-288

Author(s): Rodina L. I., Tonkov E. L.

In the terms of Lyapunov functions we obtain the conditions that allow to estimate the relative frequency of occurrence of the attainable set of a controllable system in a given set $\mathfrak{M}$. The set $\mathfrak{M}$ is called statistically invariant if the relative frequency of occurrence in $\mathfrak{M}$ is equal to one. We also derive the conditions of the statistically weak invariance of $\mathfrak{M}$ with respect to controllable system, that is, for every initial point from $\mathfrak{M}$, at least one solution of the controllable system is statistically invariant. We obtain the conditions for the attainable set to be non-wandering as well as the conditions of existence of the minimal attraction center.

Keywords: controllable systems, dynamical systems, differential inclusions, attainability, invariance, non-wandering, recurrence
Citation: Rodina L. I., Tonkov E. L., Statistical characteristics of attainable set of controllable system, non-wandering, and minimal attraction center, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 2, pp.  265-288
DOI:10.20537/nd0902008