Exponential Stability and the Markus-Yamabe Conjecture in Compact Spaces
Abstract
In this note we show that if a continuous-time, nonlinear, time-invariant, finite-dimensional system evolves on a compact subset of Rn and if the Jacobian of the vector field is Hurwitz at each point of the compact set, then there is a unique equilibrium on the set and solutions exponentially converge to it. This shows that the Markus-Yamabe conjecture, which is false in general on Rn, n>2, holds on compact sets. The results of this note can be viewed as an application of Krasovskii's method for constructing Lyapunov functions and we are able to similarly construct Lyapunov-like functions valid on the given compact set. Examples are provided to illustrate the result.
Cite
@article{arxiv.1608.08657,
title = {Exponential Stability and the Markus-Yamabe Conjecture in Compact Spaces},
author = {Ravi Mazumdar and Christopher Nielsen and Arpan Mukhopadhyay},
journal= {arXiv preprint arXiv:1608.08657},
year = {2016}
}
Comments
There is an error in the proof of Proposition II.1 which we must correct. We will re-submit once corrected