Uniform exponential stability in the sense of Hyers and Ulam for periodic time varying linear systems

Abstract

We prove that the uniform exponential stability of time depended p -periodic system Ψ̇(t) = Π(t)Ψ(t), t ∈ R+, Ψ(t) ∈ Cn is equivalent to its Hyers–Ulam stability. As a tool, we consider the exact solution of the Cauchy problem { Θ̇(t) = Π(t)Θ(t)+ eiαtζ (t), t ∈ R+ Θ(0) = Θ0 as the approximate solution of Ψ̇(t) = Π(t)Ψ(t), t ∈ R+, Ψ(t)∈ Cn , where α is any real number, ζ (t) with ζ (0) = 0 , is a p -periodic bounded function on the Banach space S (R+,C) . More precisely we prove that the system Ψ̇(t) = Π(t)Ψ(t), t ∈ R+, Ψ(t) ∈ Cn is Hyers–Ulam stable if and only if it is exponentially stable. We argue that Hyers-Ulam stability concept is quite significant in realistic problems in numerical analysis and economics.