Abstract :
Let M = {Ai} be a set of linear operators on imagen. The discrete linear inclusion DLI(M) is the set of possible trajectories (xi: i greater-or-equal, slanted 0) such that xn = AinAin−1 … Ai1x0 where image and Aij set membership, variant M. We study several notions of stability for DLI(M), including absolute asymptotic stability (AAS), which is that all products Ain … Ai1 → 0 as n → ∞. 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 image. We prove some results for AAS in this case, and give counterexamples showing that some results valid for finite sets of operators on imagen are not true for finite sets M in a general Banach algebras image.
