For switched nonlinear systems whose maps are sub-homogeneous and order-preserving on a cone, a nonlinear joint spectral radius is shown to characterize asymptotic stability and bound the worst-case separation growth rate of trajectories.
The Competive Spectral Radius of Families of Nonexpansive Mappings
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abstract
We consider a new class of repeated zero-sum games in which the payoff is the escape rate of a switched dynamical system, where at every stage, the transition is given by a nonexpansive operator depending on the actions of both players. This generalizes to the two-player (and non-linear) case the notion of joint spectral radius of a family of matrices. We show that the value of this game does exist, and we characterize it in terms of an infinite dimensional non-linear eigenproblem. This provides a two-player analogue of Ma\~ne's lemma from ergodic control. This also extends to the two-player case results of Kohlberg and Neyman (1981), Karlsson (2001), and Vigeral and the second author (2012), concerning the asymptotic behavior of nonexpansive mappings. We discuss two special cases of this game: order preserving and positively homogeneous self-maps of a cone equipped with Funk's and Thompson's metrics, and groups of translations.
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Nonlinear Joint Spectral Radius
For switched nonlinear systems whose maps are sub-homogeneous and order-preserving on a cone, a nonlinear joint spectral radius is shown to characterize asymptotic stability and bound the worst-case separation growth rate of trajectories.