TY - JOUR

T1 - Sum of Squares Certificates for Stability of Planar, Homogeneous, and Switched Systems

AU - Ahmadi, Amir Ali

AU - Parrilo, Pablo A.

N1 - Funding Information:
Manuscript received December 31, 2015; revised August 1, 2016; accepted November 21, 2016. Date of publication January 2, 2017; date of current version September 25, 2017. He is partially supported by an NSF Career Award, an AFOSR Young Investigator Program Prize, a Sloan Fellowship, and a Google Faculty Award. Recommended by Associate Editor A. Lanzon.
Publisher Copyright:
© 1963-2012 IEEE.

PY - 2017/10

Y1 - 2017/10

N2 - We show that existence of a global polynomial Lyapunov function for a homogeneous polynomial vector field or a planar polynomial vector field (under a mild condition) implies existence of a polynomial Lyapunov function that is a sum of squares (sos) and that the negative of its derivative is also a sum of squares. This result is extended to show that such sos-based certificates of stability are guaranteed to exist for all stable switched linear systems. For this class of systems, we further show that if the derivative inequality of the Lyapunov function has an sos certificate, then the Lyapunov function itself is automatically a sum of squares. These converse results establish cases where semidefinite programming is guaranteed to succeed in finding proofs of Lyapunov inequalities. Finally, we demonstrate some merits of replacing the sos requirement on a polynomial Lyapunov function with an sos requirement on its top homogeneous component. In particular, we show that this is a weaker algebraic requirement in addition to being cheaper to impose computationally.

AB - We show that existence of a global polynomial Lyapunov function for a homogeneous polynomial vector field or a planar polynomial vector field (under a mild condition) implies existence of a polynomial Lyapunov function that is a sum of squares (sos) and that the negative of its derivative is also a sum of squares. This result is extended to show that such sos-based certificates of stability are guaranteed to exist for all stable switched linear systems. For this class of systems, we further show that if the derivative inequality of the Lyapunov function has an sos certificate, then the Lyapunov function itself is automatically a sum of squares. These converse results establish cases where semidefinite programming is guaranteed to succeed in finding proofs of Lyapunov inequalities. Finally, we demonstrate some merits of replacing the sos requirement on a polynomial Lyapunov function with an sos requirement on its top homogeneous component. In particular, we show that this is a weaker algebraic requirement in addition to being cheaper to impose computationally.

KW - Asymptotic stability

KW - polynomial vector fields

KW - semidefinite programming

KW - sum of squares lyapunov functions

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U2 - 10.1109/TAC.2016.2647253

DO - 10.1109/TAC.2016.2647253

M3 - Article

AN - SCOPUS:85030985611

VL - 62

SP - 5269

EP - 5274

JO - IRE Transactions on Automatic Control

JF - IRE Transactions on Automatic Control

SN - 0018-9286

IS - 10

M1 - 7803555

ER -