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Stability of L1 contractions†

Published online by Cambridge University Press:  01 April 2015

LUIS BARREIRA  and
CLAUDIA VALLS
LUIS BARREIRA
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal. e-mail: barreira@math.ist.utl.pt; cvalls@math.ist.utl.pt
CLAUDIA VALLS
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal. e-mail: barreira@math.ist.utl.pt; cvalls@math.ist.utl.pt
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Abstract

The notion of an exponential contraction is only one among many possible rates of contraction of a nonautonomous system, while for an autonomous system all contractions are exponential. We consider the notion of an L1 contraction that includes exponential contractions as a very particular case and that is naturally adapted to the variation-of-parameters formula. Both for discrete and continuous time, we show that under very general assumptions the notion of an L1 contraction persists under sufficiently small linear and nonlinear perturbations, also maintaining the type of stability. As a natural development, we establish a version of the Grobman–Hartman theorem for nonlinear perturbations of an L1 contraction.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 1 , July 2015 , pp. 23 - 46
Copyright
Copyright © Cambridge Philosophical Society 2015 

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Footnotes

†

Supported by Portuguese funds through FCT - Fundação para a Ciência e a Tecnologia: project PEst-OE/EEI/LA0009/2013 (CAMGSD)

References

REFERENCES

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