Investigating the stability of solutions of systems of differential equations

Persidskii, S. K.

doi:10.1016/0021-8928(70)90133-4

Cited by 3 publications

(1 citation statement)

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“…On the one hand, the study of converse Lyapunov theorems has received a considerable attention from the nonlinear control community (see, for example, [18], [24], [21], [20], [22], [16] for early results). Recent works on the various variations of converse Lyapunov theorems are, among many others, [30], [14].…”

Section: Introductionmentioning

confidence: 99%

Characterizations of Global Transversal Exponential Stability

Andrieu

Jayawardhana

Praly

2021

IEEE Trans. Automat. Contr.

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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.

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