Homogeneous Lyapunov function for homogeneous continuous vector field. (English) Zbl 0762.34032

Summary: The goal of this article is to provide a construction of a homogeneous Lyapunov function \(\bar V\) associated with a system of differential equations \(\dot x=f(x)\), \(x\in\mathbb{R}^ n\) (\(n\geq 1\)), under the hypotheses: (1) \(f\in C(\mathbb{R}^ n,\mathbb{R}^ n)\) vanishes at \(x=0\) and is homogeneous; (2) the zero solution of this system is locally asymptotically stable. Moreover, the Lyapunov function \(\bar V(x)\) tends to infinity with \(\| x\|\), and belongs to \(C^ \infty(\mathbb{R}^ n\setminus\{0\},\mathbb{R})\cap C^ p(\mathbb{R}^ n,\mathbb{R})\), with \(p\in\mathbb{N}^*\) as large as wanted. As application to the theory of homogeneous systems, we present two well known results of robustness, in a slightly extended form, and with simpler proofs.


34D20 Stability of solutions to ordinary differential equations
93D09 Robust stability
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