Characterizations of global transversal exponential stability

Vincent Andrieu, Bayu Jayawardhana, Laurent Praly

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We study the relationship between the global exponential stability of an invariant manifold and the existence of a positive semi-definite Riemannian metric which is contracted by the flow. In particular, we investigate how the following properties are related to each other (in the global case): i). A manifold is globally “transversally” exponentially stable; ii). The corresponding variational system (c.f. (7) in Section II) admits the same property; iii). There exists a degenerate Riemannian metric which is contracted by the flow and can be used to construct a Lyapunov function. We show that the transverse contraction rate being larger than the expansion of the shadow on the manifold is a sufficient condition for the existence of such a Lyapunov function.
An illustration of these tools is given in the context of global full-order observer design.
Original languageEnglish
Number of pages14
JournalIEEE-Transactions on Automatic Control
Publication statusAccepted/In press - 30-Oct-2020


  • Contraction
  • Transversal exponential stability
  • Exponentially attractive invariant manifold

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