Geodesic
Statistical characteristics of attainable set of controllable system, non-wandering, and minimal attraction center
Russian journal of nonlinear dynamics, Tome 5 (2009) no. 2, pp. 265-288.

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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. 
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     author = {L. I. Rodina and E. L. Tonkov},
     title = {Statistical characteristics of attainable set of controllable system, non-wandering, and minimal attraction center},
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L. I. Rodina; E. L. Tonkov. Statistical characteristics of attainable set of controllable system, non-wandering, and minimal attraction center. Russian journal of nonlinear dynamics, Tome 5 (2009) no. 2, pp. 265-288. http://geodesic.mathdoc.fr/item/ND_2009_5_2_a7/