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Input-output conditions for the asymptotic behavior of linear skew-product flows and applications. (English) Zbl 1142.47025

Let \(\sigma\) be a discrete flow on a metric space \(\Theta\). Let \(X\) be a Banach space; denote by \({\mathcal L}(X)\) the Banach algebra of bounded linear operators on \(X\). A pair \(\pi=(\Phi,\sigma)\) is called a discrete linear skew-product flow on \(X\times\Theta\) if the mapping \(\Phi:\Theta\times{\mathbb N}\to{\mathcal L}(X)\) has the properties (i) \(\Phi(\theta,0)\) is the identity mapping and (ii) \(\Phi(\theta,m+n)=\Phi(\sigma(\theta,n),m)\Phi(\theta,n)\). Let \(\Delta({\mathbb Z},X)\) be the space of sequences \(s:{\mathbb Z}\to X\) for which the sets \(\{k\in{\mathbb Z}: s(k)\neq 0\}\) are finite. The authors give conditions under which \(\pi\) is exponentially dichotomic in terms of properties of solutions of the equation \[ \gamma(n+1)=\Phi(\sigma(\theta,n),1)\gamma(n)+s(n+1), \] where \(\theta\in\Theta\) and \(s\in\Delta({\mathbb Z},X)\).

MSC:

47D06 One-parameter semigroups and linear evolution equations
34D09 Dichotomy, trichotomy of solutions to ordinary differential equations
93D25 Input-output approaches in control theory
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
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