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Stability of discrete linear inclusion. (English) Zbl 0845.68067
Summary: Let \(M= \{A_i\}\) be a set of linear operators on \(\mathbb{R}^n\). The discrete linear inclusion DLI\((M)\) is the set of possible trajectories \((x_i: i\geq 0)\) such that \(x_n= A_{i_n} A_{i_{n- 1}}\cdots A_{i_1} x_0\), where \(x_0\in \mathbb{R}^n\) and \(A_{i_j}\in M\). We study several notions of stability for DLI\((M)\), including absolute asymptotic stability (AAS), which is that all products \(A_{i_n}\cdots A_{i_1}\to 0\) as \(n\to \infty\). We mainly study the case that \(M\) is a finite set. We give criteria for the various forms of stability. Two new approaches are taken: one relates the question of AAS of DLI\((M)\) to formal language theory and finite automata, while the second connects the AAS property to the structure of a Lie algebra associated to the elements of \(M\). More generally, the discrete linear inclusion DLI\((M)\) makes sense for \(M\) contained in a Banach algebra \(\mathcal B\).
We prove some results for AAS in this case, and give counterexamples showing that some results valid for finite sets of operators on \(\mathbb{R}^n\) are not true for finite sets \(M\) in a general Banach algebras \(\mathcal B\).

68Q45 Formal languages and automata
Full Text: DOI
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