Control systems on flag manifolds and their chain control sets

Victor Ayala; Adriano Da Silva; Luiz A. B. San Martin

doi:10.3934/dcds.2017101

Article Contents

2017, Volume 37, Issue 5: 2301-2313. Doi: 10.3934/dcds.2017101

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Control systems on flag manifolds and their chain control sets

1.
Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile and Departamento de Matemáticas Universidad Católica del Norte, Casilla 1280, Antofagasta, Chile
2.
Imecc -Unicamp, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz 13083-859, Campinas -SP, Brasil

Received: May 31, 2016

Revised: November 30, 2016

Published: May 2017

The first author was supported by Conicyt, Proyecto Fondecyt n°1150292. The second author was supported by FAPESP grants 2013/19756-8 and 2016/11135-2. The third author was supported by CNPq grant 303755/09-1, FAPESP grant 2012/18780-0, and CNPq/Universal grant 476024/2012-9.

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Abstract

A right-invariant control system $Σ$ on a connected Lie group $G$ induce affine control systems $Σ_{Θ}$ on every flag manifold $\mathbb{F}_{Θ}=G/P_{Θ}$. In this paper we show that the chain control sets of the induced systems coincides with their analogous one defined via semigroup actions. Consequently, any chain control set of the system contains a control set with nonempty interior and, if the number of the control sets with nonempty interior coincides with the number of the chain control sets, then the closure of any control set with nonempty interior is a chain control set. Some relevant examples are included.

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Mathematics Subject Classification: Primary: 93C15, 43A85; Secondary: 93D09.

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References

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