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Approximating subsystems of a dynamical system of translations. (English. Russian original) Zbl 0709.34044

Differ. Equations 25, No. 10, 1186-1195 (1989); translation from Differ. Uravn. 25, No. 10, 1705-1715 (1989).
Many applications require an analysis of the qualitative behavior of mappings of a given family \(\Lambda_*\), and of their limit properties, both when an argument tends to a limit, i.e., \(t\to \infty\), and when one tends to another, in the sense of the convergence of one mapping to another when \(t\to \infty\). We are interested in the methodology of such investigations.
This type of question is examined in the qualitative theory of differential equations [see V. V. Nemytskij and V. V. Stepanov, “Qualitative theory of differential equations.” (Russian) (1947; Zbl 0041.418) (For the 1960 English ed. see Zbl 0089.295)]. Several new problems of this kind have been encountered in optimal- control theory on large and infinite time intervals [see the author and V. I. Panasyuk, “An asymptotic magistral optimization for control systems.” (Russian) (Minsk/U.S.S.R. 1986); D. A. Carlson and A. Haurie, “Infinite horizon optimal control.” Lect. Notes Econ. Math. Syst. 290 (1987; Zbl 0649.49001)], and in set dynamics [see the author, Sib. Math. J. 27, 757-765 (1986); translation from Sib. Mat. Zh. 27, No.5(159), 155-165 (1986; Zbl 0619.34017)]. It is possible to apply to them the concepts and methods of the qualitative theory of ordinary differential equations, by using a dynamical system of translations, and approximate methods [see the author and V. I. Panasyuk, loc. cit.]. In the latter case, the mapping family \(\Lambda_*\) is approximated by mappings of a dynamical system \(\Lambda\) of translations called an approximation system, which is chosen so that the limit properties of mappings from \(\Lambda_*\) are inherited from those of elements of \(\Lambda\). In \(\Lambda\), we then find the minimal closed subsystem \(\Lambda_ 0\subset \Lambda\) inheriting the limit properties of mappings from \(\Lambda_*\), which we call the minimal approximating subsystem. By using a chain of these two approximations, we reduce the investigation of the limit properties of mappings from \(\Lambda_*\) to the investigation of limit properties of mappings from \(\Lambda_ 0\). Our objective here is to consider the questions arising in this procedure, including the determination of conditions for the existence, uniqueness, and properties of the minimal approximating subsystem \(\Lambda_ 0\), in the classical terms of minimal sets. Results in Sec. 1.5 of the joint work with V. I. Panasyuk (loc. cit) are generalized to apply to mappings with vector- valued arguments, with values in a topological space.

MSC:

37-XX Dynamical systems and ergodic theory
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