TY - JOUR
T1 - Third-order nilpotency, nice reachability and asymptotic stability
AU - Sharon, Yoav
AU - Margaliot, Michael
N1 - Funding Information:
Keywords: Nonlinear switched systems; Arbitrary switching; Global asymptotic stability; Optimal control; First-and second-order maximum principle; Bang–bang control; Singular control; Lie bracket; P. Hall basis; Sussmann’s product expansion; Chen series ✩ This work was supported by the ISF under grant number 199/03. An abridged version of this paper was presented at the 44th IEEE Conf. on Decision and Control (CDC’05). * Corresponding author. E-mail address: [email protected] (M. Margaliot). URL: http://www.eng.tau.ac.il/~michaelm (M. Margaliot).
PY - 2007/2/1
Y1 - 2007/2/1
N2 - We consider an affine control system whose vector fields span a third-order nilpotent Lie algebra. We show that the reachable set at time T using measurable controls is equivalent to the reachable set at time T using piecewise-constant controls with no more than four switches. The bound on the number of switches is uniform over any final time T. As a corollary, we derive a new sufficient condition for stability of nonlinear switched systems under arbitrary switching. This provides a partial solution to an open problem posed in [D. Liberzon, Lie algebras and stability of switched nonlinear systems, in: V. Blondel, A. Megretski (Eds.), Unsolved Problems in Mathematical Systems and Control Theory, Princeton Univ. Press, 2004, pp. 203-207].
AB - We consider an affine control system whose vector fields span a third-order nilpotent Lie algebra. We show that the reachable set at time T using measurable controls is equivalent to the reachable set at time T using piecewise-constant controls with no more than four switches. The bound on the number of switches is uniform over any final time T. As a corollary, we derive a new sufficient condition for stability of nonlinear switched systems under arbitrary switching. This provides a partial solution to an open problem posed in [D. Liberzon, Lie algebras and stability of switched nonlinear systems, in: V. Blondel, A. Megretski (Eds.), Unsolved Problems in Mathematical Systems and Control Theory, Princeton Univ. Press, 2004, pp. 203-207].
KW - Arbitrary switching
KW - Bang-bang control
KW - Chen series
KW - First- and second-order maximum principle
KW - Global asymptotic stability
KW - Lie bracket
KW - Nonlinear switched systems
KW - Optimal control
KW - P. Hall basis
KW - Singular control
KW - Sussmann's product expansion
UR - http://www.scopus.com/inward/record.url?scp=34447127569&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2006.10.011
DO - 10.1016/j.jde.2006.10.011
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AN - SCOPUS:34447127569
SN - 0022-0396
VL - 233
SP - 136
EP - 150
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 1
ER -