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On Algebraic Proofs of Stability for Homogeneous Vector Fields

Amir Ali Ahmadi (a_a_a***at***princeton.edu)
Bachir El Khadir (bkhadir***at***princeton.edu)

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.

Keywords: Converse Lyapunov theorems, nonlinear dynamics, algebraic methods in control, semidefinite programming

Category 1: Applications -- Science and Engineering (Control Applications )

Category 2: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 3: Nonlinear Optimization (Systems governed by Differential Equations Optimization )

Citation: Submitted for publication

Download: [PDF]

Entry Submitted: 03/05/2018
Entry Accepted: 03/05/2018
Entry Last Modified: 08/15/2018

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