Third-order nilpotency, finite switchings and asymptotic stability

Yoav Sharon; Michael Margaliot

doi:10.1109/CDC.2005.1583023

Third-order nilpotency, finite switchings and asymptotic stability

Yoav Sharon, Michael Margaliot^*

^*Corresponding author for this work

School of Electrical Engineering

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

Abstract

We show that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields, which span a third-order nilpotent Lie algebra, is globally asymptotically stable under arbitrary switching. This generalizes a known fact for switched linear systems and provides a partial solution to the open problem posed in [1]. To prove the result, we consider an optimal control problem which consists of finding the "most unstable" trajectory for an associated control system. We use the Agrachev-Gamkrelidze second-order maximum principle to show that there always exists an optimal control that is piecewise constant with no more than four switches. This property is obtained as a special case of a reachability result by piecewise constant controls that is of independent interest. By construction, our criterion also holds for the more general case of differential inclusions.

Original language	English
Title of host publication	Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05
Pages	5415-5420
Number of pages	6
DOIs	https://doi.org/10.1109/CDC.2005.1583023
State	Published - 2005
Event	44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05 - Seville, Spain Duration: 12 Dec 2005 → 15 Dec 2005

Publication series

Name	Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05
Volume	2005

Conference

Conference	44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05
Country/Territory	Spain
City	Seville
Period	12/12/05 → 15/12/05

Keywords

Differential inclusion
Global asymptotic stability
Lie bracket
Maximum principle
Optimal control
Switched nonlinear system

Access to Document

10.1109/CDC.2005.1583023

Cite this

Sharon, Y., & Margaliot, M. (2005). Third-order nilpotency, finite switchings and asymptotic stability. In Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05 (pp. 5415-5420). [1583023] (Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05; Vol. 2005). https://doi.org/10.1109/CDC.2005.1583023

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abstract = "We show that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields, which span a third-order nilpotent Lie algebra, is globally asymptotically stable under arbitrary switching. This generalizes a known fact for switched linear systems and provides a partial solution to the open problem posed in [1]. To prove the result, we consider an optimal control problem which consists of finding the {"}most unstable{"} trajectory for an associated control system. We use the Agrachev-Gamkrelidze second-order maximum principle to show that there always exists an optimal control that is piecewise constant with no more than four switches. This property is obtained as a special case of a reachability result by piecewise constant controls that is of independent interest. By construction, our criterion also holds for the more general case of differential inclusions.",

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Sharon, Y & Margaliot, M 2005, Third-order nilpotency, finite switchings and asymptotic stability. in Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05., 1583023, Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05, vol. 2005, pp. 5415-5420, 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05, Seville, Spain, 12/12/05. https://doi.org/10.1109/CDC.2005.1583023

Third-order nilpotency, finite switchings and asymptotic stability. / Sharon, Yoav; Margaliot, Michael.

Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05. 2005. p. 5415-5420 1583023 (Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05; Vol. 2005).

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

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N2 - We show that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields, which span a third-order nilpotent Lie algebra, is globally asymptotically stable under arbitrary switching. This generalizes a known fact for switched linear systems and provides a partial solution to the open problem posed in [1]. To prove the result, we consider an optimal control problem which consists of finding the "most unstable" trajectory for an associated control system. We use the Agrachev-Gamkrelidze second-order maximum principle to show that there always exists an optimal control that is piecewise constant with no more than four switches. This property is obtained as a special case of a reachability result by piecewise constant controls that is of independent interest. By construction, our criterion also holds for the more general case of differential inclusions.

AB - We show that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields, which span a third-order nilpotent Lie algebra, is globally asymptotically stable under arbitrary switching. This generalizes a known fact for switched linear systems and provides a partial solution to the open problem posed in [1]. To prove the result, we consider an optimal control problem which consists of finding the "most unstable" trajectory for an associated control system. We use the Agrachev-Gamkrelidze second-order maximum principle to show that there always exists an optimal control that is piecewise constant with no more than four switches. This property is obtained as a special case of a reachability result by piecewise constant controls that is of independent interest. By construction, our criterion also holds for the more general case of differential inclusions.

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KW - Global asymptotic stability

KW - Lie bracket

KW - Maximum principle

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Sharon Y, Margaliot M. Third-order nilpotency, finite switchings and asymptotic stability. In Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05. 2005. p. 5415-5420. 1583023. (Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05). doi: 10.1109/CDC.2005.1583023

Third-order nilpotency, finite switchings and asymptotic stability

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