TY - GEN
T1 - On higher order derivatives of Lyapunov functions
AU - Ahmadi, Amir Ali
AU - Parrilo, Pablo A.
PY - 2011/9/29
Y1 - 2011/9/29
N2 - This note is concerned with a class of differential inequalities in the literature that involve higher order derivatives of Lyapunov functions and have been proposed to infer asymptotic stability of a dynamical system without requiring the first derivative of the Lyapunov function to be negative definite. We show that whenever a Lyapunov function satisfies these conditions, we can explicitly construct another (standard) Lyapunov function that is positive definite and has a negative definite first derivative. Our observation shows that a search for a standard Lyapunov function parameterized by higher order derivatives of the vector field is less conservative than the previously proposed conditions. Moreover, unlike the previous inequalities, the new inequality can be checked with a convex program. This is illustrated with an example where sum of squares optimization is used.
AB - This note is concerned with a class of differential inequalities in the literature that involve higher order derivatives of Lyapunov functions and have been proposed to infer asymptotic stability of a dynamical system without requiring the first derivative of the Lyapunov function to be negative definite. We show that whenever a Lyapunov function satisfies these conditions, we can explicitly construct another (standard) Lyapunov function that is positive definite and has a negative definite first derivative. Our observation shows that a search for a standard Lyapunov function parameterized by higher order derivatives of the vector field is less conservative than the previously proposed conditions. Moreover, unlike the previous inequalities, the new inequality can be checked with a convex program. This is illustrated with an example where sum of squares optimization is used.
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M3 - Conference contribution
AN - SCOPUS:80053157136
SN - 9781457700804
T3 - Proceedings of the American Control Conference
SP - 1313
EP - 1314
BT - Proceedings of the 2011 American Control Conference, ACC 2011
T2 - 2011 American Control Conference, ACC 2011
Y2 - 29 June 2011 through 1 July 2011
ER -