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In this paper we study the controllability property of invariant control systems on Lie groups. In [1], the authors state: ``If there exists a real function strictly increasing on the positive trajectories, then the system cannot be controllable". To develop this idea, the authors define the concept of symplectic vector via the co-adjoint representation. We are interested in finding algebraic conditions to determine the existence of symplectic vectors in nilpotent Lie algebras. In particular, we...
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