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Statistical cluster points and turnpike. (English) Zbl 0963.40002
Summary: We study an asymptotic behaviour of optimal paths of a difference inclusion. Some definitions of the turnpike property [see, e.g., {\it V. L. Makarov}, {\it A. M. Rubinov} and {\it M. J. Levin}, Mathematical economic theory: pure and mixed types of economic mechanisms (Advanced Textbooks in Economics. 33. Amsterdam: North-Holland) (1995; Zbl 0834.90001)] assume that there is a certain stationary point and optimal paths converge to that point. In this case only a finite number of terms of the path (sequence) remain outside of every neighbourhood of that point. In the present paper a statistical cluster point introduced in {\it J. A. Fridy} [Proc. Am. Math. Soc. 118, No. 4, 1187--1192 (1993; Zbl 0776.40001)] 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 all optimal paths. In this case not only a finite number but also an infinite number of terms of the path may remain 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 much weaker than the usual convexity assumption. These assumptions were first introduced for continuous systems in {\it M. A. Mamedov} [Optimizatsiya 36(53), 101--112 (1985; Zbl 0614.90023)].

##### MSC:
 40A05 Convergence and divergence of series and sequences 49J24 Optimal control problems with differential inclusions (existence) (MSC2000) 49K40 Sensitivity, stability, well-posedness of optimal solutions
##### Keywords:
turnpike; statistical cluster point
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