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# Statistical cluster points and turnpike

Version 2 2024-06-05, 03:23

Version 1 2019-07-18, 15:09

journal contribution

posted on 2024-06-05, 03:23 authored by S Pehlivan, MA MamedovIn this paper we study an asymptotic behaviour of optimal paths of a difference inclusion. The turnpike property in some wording [5,8, and so on] provided that there is a certain stationary point and optimal paths converge to that point. In this case only a finite number terms of the path (sequence) remain on the outside of every neighbourhood of that point. In the present paper a statistical cluster point introduced in [1] instead of the usual concept of limit point is considered and the turnpike theorem is proved. Here it is established that there exists a stationary point which is a statistical cluster point for the all optimal paths. In this case not only a finite number but also infinite number terms of the path may remain on the outside of every small neighbourhood of the stationary point, but the number of these terms in comparison with the number of terms in the neighbourhood is so small that we can say: the path "almost" remains in this neighbourhood. Note that the main results are obtained under certain assumptions which are essentially weaker than the usual convexity assumption. These assumptions first were introduced for continuous systems in [6].