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Ψ-uniformly stability for time varying linear dynamic systems on time scales. (English) Zbl 1151.34043

A linear dynamic equation

x Δ =A(t)x

is said to be Ψ-uniformly stable, provided there exists a γ>0 and diagonal matrices Ψ(t) (with positive diagonal elements) such that the transition matrix Φ A (t,s) satisfies

Ψ(t)Φ A (t,s)Ψ(s) -1 γforallst·

It is shown that Ψ-uniform stability persists under linear-homogeneous and linear-inhomogeneous perturbations, provided their L 1 -norm is appropriately weighted.

34D20Stability of ODE
34D10Stability perturbations of ODE
34A30Linear ODE and systems, general
39A10Additive difference equations
39A11Stability of difference equations (MSC2000)
39A12Discrete version of topics in analysis