Summary: ESAIM: COCV ESAIM: Control, Optimisation and Calculus of Variations
July 2006, Vol. 12, 409441 www.edpsciences.org/cocv
AN ESTIMATION OF THE CONTROLLABILITY TIME FOR SINGLE-INPUT
SYSTEMS ON COMPACT LIE GROUPS
and Thomas Chambrion2
Abstract. Geometric control theory and Riemannian techniques are used to describe the reachable
set at time t of left invariant single-input control systems on semi-simple compact Lie groups and to
estimate the minimal time needed to reach any point from identity. This method provides an effective
way to give an upper and a lower bound for the minimal time needed to transfer a controlled quantum
system with a drift from a given initial position to a given final position. The bounds include diameters
of the flag manifolds; the latter are also explicitly computed in the paper.
Mathematics Subject Classification. 22E46, 93B03.
Received September 4, 2004. Revised May 8, 2005.
1.1. Physical context
Some recent papers [912, 14, 15, 22, 32] have studied the problem of transferring a population of quantum
particles from a quantum state to another, using an external field (laser). In many applications (e.g., MNR)
a crucial issue is to achieve this quantum transfer in the shortest possible time, in order to avoid relaxation