Stability of nonautonomous equations and Lyapunov functions

Luis Barreira; Claudia Valls

doi:10.3934/dcds.2013.33.2631

Article Contents

2013, Volume 33, Issue 7: 2631-2650. Doi: 10.3934/dcds.2013.33.2631

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Stability of nonautonomous equations and Lyapunov functions

Luis Barreira^1, and
Claudia Valls^2,

1.
Departamento de Matemática, Instituto Superior Técnico, UTL, 1049-001 Lisboa
2.
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa

Received: April 30, 2011

Revised: October 31, 2012

Published: June 2013

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Abstract

We consider nonautonomous linear equations $x'=A(t)x$ in a Banach space, and we give a complete characterization of those admitting nonuniform exponential contractions in terms of strict Lyapunov functions. The uniform contractions are a very particular case of nonuniform exponential contractions. In addition, we establish ``inverse theorems'' that give explicitly a strict Lyapunov function for each nonuniform contraction. These functions are constructed in terms of Lyapunov norms, which transform the nonuniform behavior of the contraction into a uniform exponential behavior. Moreover, we use the characterization of nonuniform exponential contractions in terms of strict Lyapunov functions to establish in a very simple manner, in comparison with former works, the persistence of the asymptotic stability under sufficiently small linear and nonlinear perturbations.

Keywords:

Lyapunov functions,
nonuniform exponential contractions.

Mathematics Subject Classification: Primary: 37D99, 93D99.

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References

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