A necessary condition for quantitative exponential stability of linear state-space systems

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Abstract

For an arbitrary n × n constant matrix A the two following facts are well known: • (1/n)Re(trace A) - maxj=i,...,nRe λj(A4)≤0; • If U is a unitary matrix, one can always find a skew-Hermitian matrix A so that U = eA. In this note we present the extension of these two facts to the context of linear time-varying dynamical systems ẋ = A(t)x, t≥0. As a by-product, this result suggests that, the notion of "slowly varying state-space systems", commonly used in literature, is mathematically not natural to the problem of exponential stability.

Original languageEnglish
Pages (from-to)1-4
Number of pages4
JournalSystems and Control Letters
Volume38
Issue number1
DOIs
StatePublished - 15 Sep 1999

Keywords

  • Convergence rate
  • Exponential stability
  • Necessary condition
  • Slowly time-varying

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