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September  2002, 1(3): 285-312. doi: 10.3934/cpaa.2002.1.285

On the minimum time problem for driftless left-invariant control systems on SO(3)

Ugo Boscain 1, and  Yacine Chitour 2,

1. 

Universite de Bourgogne, Departement de Mathematiques, Analyse Appliquee et Optimisation, 47870-21078 Dijon, France
2. 

Universite Paris XI,, Departement de Mathematiques, F-91405 Orsay, France

Received  July 2001 Revised  January 2002 Published  June 2002

In this paper, we investigate the structure of time-optimal trajectories for a driftless control system on $SO(3)$ of the type $\dot x=x(u_1f_1+u_2f_2), \quad |u_1|, \quad |u_2|\leq 1$, where $f_1,\quad f_2\in so(3)$ define two linearly independent left-invariant vector fields on $SO(3)$. We show that every time-optimal trajectory is a finite concatenation of at most five (bang or singular) arcs. More precisely, a time-optimal trajectory is, on the one hand, bang-bang with at most either two consecutive switchings relative to the same input or three switchings alternating between two inputs, or, on the other hand, a concatenation of at most two bangs followed by a singular arc and then two other bangs. We end up finding a finite number of three-parameters trajectory types that are sufficient for time-optimality.
Keywords: Pontryagin maximum principle, Time-optimal problems, left-invariant control systems on SO(3), envelope theory..
Mathematics Subject Classification: 49K15, 49J1.
Citation: Ugo Boscain, Yacine Chitour. On the minimum time problem for driftless left-invariant control systems on SO(3). Communications on Pure & Applied Analysis, 2002, 1 (3) : 285-312. doi: 10.3934/cpaa.2002.1.285
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