Existence of piecewise affine Lyapunov functions in two dimensions (Q986588)

The authors consider the problem of existence of piecewise affine Lyapunov functions for autonomous systems of the form \[ \dot x=f(x) \] under the assumptions that \(f\in{C}^{1}\left(\mathbb{R}^{n},\mathbb{R}^{n}\right)\) and that the origin is an exponentially stable equilibrium of the system. The main result of the paper is a theorem which states, that in the case \(n=2\), there is a local triangulation scheme such that the system possesses a piecewise affine Lyapunov function in a neighborhood of the equilibrium.