Exponential Stability and the Markus-Yamabe Conjecture in Compact Spaces

TL;DR

This paper proves that in compact spaces, a nonlinear system with a Hurwitz Jacobian at every point has a unique equilibrium and solutions exponentially converge to it, confirming the Markus-Yamabe conjecture in this setting.

Contribution

It demonstrates that the Markus-Yamabe conjecture holds on compact sets for nonlinear systems with Hurwitz Jacobians, extending known results.

Findings

Unique equilibrium exists on compact sets with Hurwitz Jacobian.

Solutions exponentially converge to the equilibrium.

The conjecture holds in compact spaces, contrary to the general case in Rn.

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.