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
SN - 0018-9286
VL - 62
SP - 5269
EP - 5274
JO - IRE Transactions on Automatic Control
JF - IRE Transactions on Automatic Control
IS - 10
M1 - 7803555
ER -