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Characterizations of Global Transversal Exponential Stability

Abstract : We study the relationship between the global exponential stability of an invariant manifold and the existence of a semi-definite positive 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. (8) 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.
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Submitted on : Wednesday, December 22, 2021 - 1:53:12 PM
Last modification on : Tuesday, January 4, 2022 - 5:58:11 AM

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Vincent Andrieu, Bayu Jayawardhana, Laurent Praly. Characterizations of Global Transversal Exponential Stability. IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2021, 66 (8), pp.3682-3694. ⟨10.1109/TAC.2020.3036021⟩. ⟨hal-03500595⟩

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