Abstract
Like in the classical linear Euclidean system, we would like to characterize for a linear control system on a connected Lie group G its control set with nonempty interior that contains the identity of G. We show that many topological properties of this control set are intrinsically connected with the eigenvalues of a derivation associated to the drift of the system. In particular, we prove that if G is a decomposable Lie group there exists only one control set with nonempty interior for the whole linear system. Furthermore, for nilpotent Lie groups we characterize when this set is bounded.
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Acknowledgements
We would like to thanks the Centro de Estudios Científicos (CECs) in Valdivia, Chile, through Prof. Dr. Jorge Zanelli, to provide to the authors an excellent environment to work out on this article. V. Ayala supported by Proyecto Fondecyt no. 1150292. Conicyt, Chile. A. Da Silva supported by Fapesp Grant no. 2013/19756-8. G. Zsigmond supported by CAPES Grant BEX 1041-14-2.
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Ayala, V., Da Silva, A. & Zsigmond, G. Control sets of linear systems on Lie groups. Nonlinear Differ. Equ. Appl. 24, 8 (2017). https://doi.org/10.1007/s00030-017-0430-5
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DOI: https://doi.org/10.1007/s00030-017-0430-5