Controllability of invariant control systems at uniform time

Let G be a compact and connected semisimple Lie group and Σ an invariant control systems on G. Our aim in this work is to give a new proof of Theorem 1 proved by Jurdjevic and Sussmann in [6]. Precisely, to find a positive time S_Σ such that the system turns out controllable at uniform time S_Σ. Our proof is different, elementary and the main argument comes directly from the definition of semisimple Lie group. The uniform time is not arbitrary. Finally, if A = ∪_t>0 A(t, e) denotes the reachable set from arbitrary uniform time, we conjecture that it is possible to determine A as the intersection of the isotropy groups of orbits of G-representations which contains exp(Z), where Z is the Lie algebra determined by the control vectors.