## 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.

### MSC:

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

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