A Poincar\'e-Bendixson theorem for Bebutov shifts and applications to switched systems

TL;DR

This paper extends the Poincaré-Bendixson theorem to certain curves on the 2-sphere, providing new criteria for periodic trajectories and stability in low-dimensional switched systems, resolving longstanding conjectures in control theory.

Contribution

It introduces a novel version of the Poincaré-Bendixson theorem for curves on the 2-sphere and applies it to establish stability and periodicity conditions in low-dimensional switched systems.

Findings

Periodic asymptotic stability implies global exponential stability in 3D real linear switched systems.

New criteria for the existence of periodic trajectories in affine control systems.

Resolution of a continuous-time analogue of the Lagarias-Wang finiteness conjecture.

Abstract

We prove a version of the Poincar\'e-Bendixson theorem for certain classes of curves on the 2-sphere which are not required to be the trajectories of an underlying flow or semiflow on the sphere itself. Using this result we extend the Poincar\'e-Bendixson theorem to the context of continuous semiflows on compact subsets of the 2-sphere and the projective plane, give new sufficient conditions for the existence of periodic trajectories of certain low-dimensional affine control systems, and give a new criterion for the global uniform exponential stability of switched systems of homogeneous ODEs in dimension three. We prove in particular that periodic asymptotic stability implies global uniform exponential stability for real linear switched systems of dimension three and complex linear switched systems of dimension two. In combination with a recent result of the second author, this resolves a question of R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King and resolves a natural analogue of the Lagarias-Wang finiteness conjecture in continuous time.