Geodesic
On the equivalence of control systems on Lie groups
Communications in Mathematics, Tome 23 (2015) no. 2, pp. 119-129.

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We consider state space equivalence and feedback equivalence in the context of (full-rank) left-invariant control systems on Lie groups. We prove that two systems are state space equivalent (resp.~detached feedback equivalent) if and only if there exists a Lie group isomorphism relating their parametrization maps (resp. traces). Local analogues of these results, in terms of Lie algebra isomorphisms, are also found. Three illustrative examples are provided.
Classification : 22E60, 93B27
Keywords: left-invariant control system; state space equivalence; detached feedback equivalence 
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Biggs, Rory; Remsing, Claudiu C. On the equivalence of control systems on Lie groups. Communications in Mathematics, Tome 23 (2015) no. 2, pp. 119-129. http://geodesic.mathdoc.fr/item/COMIM_2015__23_2_a2/