TY - JOUR

T1 - On Algebraic Proofs of Stability for Homogeneous Vector Fields

AU - Ahmadi, Amir Ali

AU - El Khadir, Bachir

N1 - Funding Information:
Manuscript received March 5, 2018; revised March 9, 2018, August 19, 2018, and March 29, 2019; accepted April 16, 2019. Date of publication May 6, 2019; date of current version December 27, 2019. This work was supported in part by the DARPA Young Faculty Award, in part by the MURI Award of the AFOSR, in part by the CAREER Award of the NSF, in part by the Innovation Fund of the School of Engineering at Princeton University, in part by the Google Faculty Award, and in part by the Sloan Fellowship. Recommended by Associate Editor C. W. Scherer. (Corresponding author: Amir Ali Ahmadi.) The authors are with the Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544 USA (e-mail:,a_a_a@princeton.edu; bkhadir@princeton.edu).
Publisher Copyright:
© 1963-2012 IEEE.

PY - 2020/1

Y1 - 2020/1

N2 - 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 nonhomogeneous systems by proving asymptotic stability of their lowest order homogeneous component. This paper also includes some negative results: We show that in absence of homogeneity, globally asymptotically stable polynomial vector fields may fail to admit a global rational Lyapunov function, and 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.

AB - 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 nonhomogeneous systems by proving asymptotic stability of their lowest order homogeneous component. This paper also includes some negative results: We show that in absence of homogeneity, globally asymptotically stable polynomial vector fields may fail to admit a global rational Lyapunov function, and 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.

KW - Algebraic methods in control

KW - converse lyapunov theorems

KW - nonlinear dynamics

KW - rational Lyapunov functions

KW - semidefinite programming

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U2 - 10.1109/TAC.2019.2914968

DO - 10.1109/TAC.2019.2914968

M3 - Article

AN - SCOPUS:85077799419

SN - 0018-9286

VL - 65

SP - 325

EP - 332

JO - IRE Transactions on Automatic Control

JF - IRE Transactions on Automatic Control

IS - 1

M1 - 8706528

ER -