The Lp-version of the generalised Bohl-Perron principle for vector equations with delay

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Abstract

We consider the equation ẏ = Ey, where Ey(t) = ∫ d τR (t,τ)y(t - τ) (t ≥ 0) with an n x n-matrix-valued function R(t,τ). It is proved that, if for a p ≥ 1, the non-homogeneous equation ẋ = Ex + f with the zero initial condition, for any f ∈ Lp, has a solution x ∈ Lp, then the considered homogeneous equation is exponentially stable. By that result, sharp stability conditions are derived for vector functional differential equations 'close' to autonomous ones and for equations with small delays.

Original languageEnglish
Pages (from-to)448-458
Number of pages11
JournalInternational Journal of Dynamical Systems and Differential Equations
Volume3
Issue number4
DOIs
StatePublished - 1 Jan 2011

Keywords

  • Exponential stability
  • Functional differential equation
  • Linear equation

ASJC Scopus subject areas

  • General Engineering
  • Discrete Mathematics and Combinatorics
  • Control and Optimization

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