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Indefinability in o-minimal structures of finite sets of matrices whose infinite products converge and are bounded or unbounded. (English. Russian original) Zbl 1078.93017

Autom. Remote Control 64, No. 9, 1386-1400 (2003); translation from Avtom. Telemekh. 2003, No. 9, 24-41 (2003).
Summary: This paper is concerned with the convergence and boundedness or unboundedness of the set of all possible matrix products with coefficients belonging to some finite set, i.e., the problem to which many problems of control theory and mathematics are reduced. The indefinability of this problem in o-minimal structures containing semialgebraic sets, which can be regarded as characteristic for the complexity of the problem, is demonstrated. The result shows, in particular, that the solution of our problem cannot be found as a finite Boolean combination of conditions containing a finite number of ordinary arithmetical operations of addition, subtraction, and multiplication, as well as exponentiation and application of bounded analytic functions.

MSC:

93B27 Geometric methods
03C64 Model theory of ordered structures; o-minimality
14P10 Semialgebraic sets and related spaces
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