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07 February 2025
 
  » arxiv » 1608.8657

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Exponential Stability and the Markus-Yamabe Conjecture in Compact Spaces
Ravi Mazumdar ; Christopher Nielsen ; Arpan Mukhopadhyay ;
Date 2 Sep 2016
AbstractIn 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.
Source arXiv, 1608.8657
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