TY - GEN
T1 - Third-order nilpotency, finite switchings and asymptotic stability
AU - Sharon, Yoav
AU - Margaliot, Michael
PY - 2005
Y1 - 2005
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.
KW - Differential inclusion
KW - Global asymptotic stability
KW - Lie bracket
KW - Maximum principle
KW - Optimal control
KW - Switched nonlinear system
UR - http://www.scopus.com/inward/record.url?scp=33847238017&partnerID=8YFLogxK
U2 - 10.1109/CDC.2005.1583023
DO - 10.1109/CDC.2005.1583023
M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???
AN - SCOPUS:33847238017
SN - 0780395689
SN - 9780780395688
T3 - Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05
SP - 5415
EP - 5420
BT - Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05
T2 - 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05
Y2 - 12 December 2005 through 15 December 2005
ER -