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Keywords:
uniform-time; compact; semisimple; reverse-system
Summary:
Let \$G\$ be a compact and connected semisimple Lie group and \$\Sigma \$ 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_{\Sigma }\$ such that the system turns out controllable at uniform time \$s_{\Sigma }\$. 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=\bigcap _{ 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 (\mathfrak{z})\$, where \$\mathfrak{z}\$ is the Lie algebra determined by the control vectors.
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