Characterizations of global transversal exponential stability (long version)

Vincent Andrieu (1) , Bayu Jayawardhana (2) , Laurent Praly (3)

1 LAGEPP - Laboratoire d'automatique, de génie des procédés et de génie pharmaceutique
2 ITM
3 CAS - Centre Automatique et Systèmes

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.

Mots clés: transversal exponential stability; exponentially attractive invariant manifold; Contraction; ex- ponentially attractive invariant manifold

Domaines: Automatique / Robotique; Optimisation et contrôle [math.OC]

Vincent Andrieu, Bayu Jayawardhana, Laurent Praly. Characterizations of global transversal exponential stability (long version). 2020. ⟨hal-02851212v2⟩