Abstract
This paper is devoted to the study of controllability of linear systems on solvable and nilpotent Lie groups. Some general results are stated and used to completely characterize the controllable systems on the nilpotent Heisenberg group and the solvable two-dimensional affine group.
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Notes
A Lie algebra \(\mathfrak {g}\) is solvable if its derived series terminates in the zero Lie algebra: define \(\mathcal {D}^{1}\mathfrak {g}=[\mathfrak {g},\mathfrak {g}]\) and by induction \(\mathcal {D}^{n+1}\mathfrak {g}=[\mathcal {D}^{n}\mathfrak {g},\mathcal {D}^{n}\mathfrak {g}]\), then \(\mathfrak {g}\) is solvable if \(\mathcal {D}^{n}\mathfrak {g}\) vanishes for some integer n.
The Lie bracket of \(\mathfrak {g}\) is chosen to be equal to the vector fields bracket. For this reason its sign for right-invariant vector fields is not the same as for left-invariant ones.
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Acknowledgments
The authors wish to express their thanks to Saïd Naciri for pointing an error in the proof of Theorem 4.
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Dath, M., Jouan, P. Controllability of Linear Systems on Low Dimensional Nilpotent and Solvable Lie Groups. J Dyn Control Syst 22, 207–225 (2016). https://doi.org/10.1007/s10883-014-9258-z
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DOI: https://doi.org/10.1007/s10883-014-9258-z