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 -