On Algebraic Proofs of Stability for Homogeneous Vector Fields

TL;DR

This paper proves that asymptotically stable homogeneous vector fields admit rational Lyapunov functions, and explores the algebraic and computational aspects of constructing such functions, including limitations and advantages over polynomial Lyapunov functions.

Contribution

It establishes the existence of rational Lyapunov functions for stable homogeneous vector fields and connects sum of squares certificates with semidefinite programming, extending classical stability results.

Findings

Stable homogeneous vector fields have rational Lyapunov functions.

Sum of squares certificates enable semidefinite programming for polynomial cases.

Limitations include non-existence of global rational Lyapunov functions without homogeneity.

Abstract

We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the rational function and its derivative have sum of squares certificates and hence such a Lyapunov function can always be found by semidefinite programming. This generalizes the classical fact that an asymptotically stable linear system admits a quadratic Lyapunov function which satisfies a certain linear matrix inequality. In addition to homogeneous vector fields, the result can be useful for showing local asymptotic stability of non-homogeneous systems by proving asymptotic stability of their lowest order homogeneous component. This paper also includes some negative results: We show that (i) in absence of homogeneity, globally asymptotically stable polynomial vector fields may fail to admit a global rational Lyapunov function, and (ii) in presence of homogeneity, the degree of the numerator of a rational Lyapunov function may need to be arbitrarily high (even for vector fields of fixed degree and dimension). On the other hand, we also give a family of homogeneous polynomial vector fields that admit a low-degree rational Lyapunov function but necessitate polynomial Lyapunov functions of arbitrarily high degree. This shows the potential benefits of working with rational Lyapunov functions, particularly as the ones whose existence we guarantee have structured denominators and are not more expensive to search for than polynomial ones.