2016/08/30 by Ravi R. Mazumdar, Mazumdar, Ravi, Christopher Nielsen +3
Engineering · #Control and Stability of Dynamical Systems #Dynamical Systems (math.DS) #Dynamics and Control of Mechanical Systems #FOS: Mathematics #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1608.08657
openalex publication_date 2016/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.