Mathematics

Lie group formalism and riemanian techniques can be used to get an estimation of the diameter of SU(n) under a single input left invariant control system, that is to say a system with a drift and one unbounded real control. The first step is to apply the technique of variation of the constant to approximate the reachable set at time T of the initial system. The second step consits in the replacement of the initial system by a very close Riemanian one. Then, using the particular structure of the Lie algebra su(n), it is possible to use well-known riemanian results to get an upper bound for the diameter of the group. Although the first steps are valid in any semi-simple compact real Lie group, the result will be presented only in the case of the special unitary group SU(n), in order to avoid technical considerations on general Lie groups.