A generalized chain rule and a bound on the continuity of solutions and converse Lyapunov functions

Matthew M. Peet

doi:10.1109/CDC.2009.5400414

A generalized chain rule and a bound on the continuity of solutions and converse Lyapunov functions

Matthew M. Peet

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

2 Scopus citations

Abstract

This paper gives a bound on the continuity of solutions to nonlinear ordinary differential equations. Continuity is measured with respect to an arbitrary Sobolev norm. This result is used to give a bound on the continuity of a common converse Lyapunov function. A major technical contribution of this paper is to give an explicit formula for n^th-degree derivatives of the composition of differentiable mappings from ℝⁿ to ℝⁿ. This is a generalization of the formula of Faa di Bruno which dealt with differentiable mappings from ℝ to ℝ. It is expected that continuity bounds of the type given in this paper can be used to prove the existence of bounded-degree polynomial Lyapunov functions or give bounds on the Lyapunov exponent.

Original language	English (US)
Title of host publication	Proceedings of the 48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009
Publisher	Institute of Electrical and Electronics Engineers Inc.
Pages	3155-3161
Number of pages	7
ISBN (Print)	9781424438716
DOIs	https://doi.org/10.1109/CDC.2009.5400414
State	Published - 2009
Externally published	Yes
Event	48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009 - Shanghai, China Duration: Dec 15 2009 → Dec 18 2009

Publication series

Name	Proceedings of the IEEE Conference on Decision and Control
ISSN (Print)	0743-1546
ISSN (Electronic)	2576-2370

Other

Other	48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009
Country/Territory	China
City	Shanghai
Period	12/15/09 → 12/18/09

ASJC Scopus subject areas

Control and Systems Engineering
Modeling and Simulation
Control and Optimization

Access to Document

10.1109/CDC.2009.5400414

Cite this

Peet, M. M. (2009). A generalized chain rule and a bound on the continuity of solutions and converse Lyapunov functions. In Proceedings of the 48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009 (pp. 3155-3161). Article 5400414 (Proceedings of the IEEE Conference on Decision and Control). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/CDC.2009.5400414

A generalized chain rule and a bound on the continuity of solutions and converse Lyapunov functions. / Peet, Matthew M.
Proceedings of the 48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009. Institute of Electrical and Electronics Engineers Inc., 2009. p. 3155-3161 5400414 (Proceedings of the IEEE Conference on Decision and Control).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Peet, MM 2009, A generalized chain rule and a bound on the continuity of solutions and converse Lyapunov functions. in Proceedings of the 48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009., 5400414, Proceedings of the IEEE Conference on Decision and Control, Institute of Electrical and Electronics Engineers Inc., pp. 3155-3161, 48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009, Shanghai, China, 12/15/09. https://doi.org/10.1109/CDC.2009.5400414

Peet MM. A generalized chain rule and a bound on the continuity of solutions and converse Lyapunov functions. In Proceedings of the 48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009. Institute of Electrical and Electronics Engineers Inc. 2009. p. 3155-3161. 5400414. (Proceedings of the IEEE Conference on Decision and Control). doi: 10.1109/CDC.2009.5400414

Peet, Matthew M. / A generalized chain rule and a bound on the continuity of solutions and converse Lyapunov functions. Proceedings of the 48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009. Institute of Electrical and Electronics Engineers Inc., 2009. pp. 3155-3161 (Proceedings of the IEEE Conference on Decision and Control).

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A generalized chain rule and a bound on the continuity of solutions and converse Lyapunov functions

Abstract

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