Non-Sturmian sequences of matrices providing the maximum growth rate of matrix products

In the theory of linear switching systems with discrete time, as in other areas of mathematics, the problem of studying the growth rate of the norms of all possible matrix products Aσn⋯Aσ0 with factors from a set of matrices A arises. So far, only for a relatively small number of classes of matrices...

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Vydáno v:Automatica (Oxford) Ročník 145; s. 110574
Hlavní autor: Kozyakin, Victor
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier Ltd 01.11.2022
Témata:
Barabanov norm
Growth rate
Infinite matrix products
Linear switching systems
Python program
Sturmian sequences
Python program
Sturmian sequences
Growth rate
Barabanov norm
Infinite matrix products
Linear switching systems
ISSN:0005-1098, 1873-2836
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Popis
Shrnutí:In the theory of linear switching systems with discrete time, as in other areas of mathematics, the problem of studying the growth rate of the norms of all possible matrix products Aσn⋯Aσ0 with factors from a set of matrices A arises. So far, only for a relatively small number of classes of matrices A has it been possible to accurately describe the sequences of matrices that guarantee the maximum rate of increase of the corresponding norms. Moreover, in almost all cases studied theoretically, the index sequences {σn} of matrices maximizing the norms of the corresponding matrix products have been shown to be periodic or so-called Sturmian, which entails a whole set of “good” properties of the sequences {Aσn}, in particular the existence of a limiting frequency of occurrence of each matrix factor Ai∈A in them. In the paper it is shown that this is not always the case: a class of matrices is defined consisting of two 2×2 matrices, similar to rotations in the plane, in which the sequence {Aσn} maximizing the growth rate of the norms ‖Aσn⋯Aσ0‖ is not Sturmian. All considerations are based on numerical modeling and cannot be considered mathematically rigorous in this part; rather, they should be interpreted as a set of questions for further comprehensive theoretical analysis.
ISSN:0005-1098
1873-2836
DOI:10.1016/j.automatica.2022.110574