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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 $$\mathcal B$$.

##### MSC:
 68Q45 Formal languages and automata
##### Keywords:
formal language theory; finite automata
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##### References:
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