May  2017, 37(5): 2301-2313. doi: 10.3934/dcds.2017101

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  June 2016 Revised  December 2016 Published  February 2017

Fund Project: 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

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.

Citation: Victor Ayala, Adriano Da Silva, Luiz A. B. San Martin. Control systems on flag manifolds and their chain control sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2301-2313. doi: 10.3934/dcds.2017101
References:
[1]

V. Ayala and L. A. B. San Martin, Controllability of two-dimensional bilinear control systems: Restricted Controls and discrete-time, Proyecciones, 18 (1999), 207-223. Google Scholar

[2]

C. J. Braga Barros and L. A. B. San Martin, Chain control sets for semigroup actions, Mat. Apl. Comp., 15 (1996), 257-276. Google Scholar

[3]

C. J. Braga Barros and L. A. B. San Martin, Chain transitive sets for flows on flag bundles, Forum Math., 19 (2007), 19-60. doi: 10.1515/FORUM.2007.002. Google Scholar

[4]

R. W. Brocke, System theory on group manifolds and coset spaces, SIAM Journal on Control, 10 (1972), 265-284. Google Scholar

[5]

F. Colonius and W. Kliemann, The Dynamics of Control Birkhauser, 2000. doi: 10.1007/978-1-4612-1350-5. Google Scholar

[6]

A. Da Silva and C. Kawan, Hyperbolic chain control sets on flag manifolds, Journal of Dynamics and Control Systems, 22 (2016), 725-745. doi: 10.1007/s10883-015-9308-1. Google Scholar

[7]

A. Da Silva and C. Kawan, Invariance entropy of hyperbolic control sets, Discrete and Continuous Dynamical Systems, 36 (2016), 97-136. doi: 10.3934/dcds.2016.36.97. Google Scholar

[8]

V. Jurjevic, Geometric Control Theory Cambridge University Press, 1997.Google Scholar

[9]

V. Jurjevic and H. Sussmann, Control systems on Lie groups, Journal of Differential Equations, 12 (1972), 313-329. Google Scholar

[10]

C. Kawan, Invariance Entropy for Deterministic Control SystemsAn Introduction, Lecture Notes in Mathematics, 2089, Springer-Verlag, Berlin, 2013.Google Scholar

[11]

M. Patrao and L.A.B. San Martin, Semiflows on topological spaces: Chain transitivity and semigroups, Journal of Dynamics and Differential Equations, 19 (2007), 155-180. Google Scholar

[12]

L. A. B. San Martin, Order and domains of attractions of control sets in flag manifolds, Journal of Lie theory, 8 (1998), 335-350. Google Scholar

[13]

L. A. B. San Martin and L. Seco, Morse and Lyapunov spectra and dynamics on flag bundles, Ergod. Th. & Dynam. Sys., 30 (2010), 893-922. doi: 10.1017/S0143385709000285. Google Scholar

[14]

L. A. B. San Martin and P. A. Tonelli, Semigroup actions on Homogeneous Spaces, Semigroup Forum, 50 (1995), 59-88. doi: 10.1007/BF02573505. Google Scholar

show all references

References:
[1]

V. Ayala and L. A. B. San Martin, Controllability of two-dimensional bilinear control systems: Restricted Controls and discrete-time, Proyecciones, 18 (1999), 207-223. Google Scholar

[2]

C. J. Braga Barros and L. A. B. San Martin, Chain control sets for semigroup actions, Mat. Apl. Comp., 15 (1996), 257-276. Google Scholar

[3]

C. J. Braga Barros and L. A. B. San Martin, Chain transitive sets for flows on flag bundles, Forum Math., 19 (2007), 19-60. doi: 10.1515/FORUM.2007.002. Google Scholar

[4]

R. W. Brocke, System theory on group manifolds and coset spaces, SIAM Journal on Control, 10 (1972), 265-284. Google Scholar

[5]

F. Colonius and W. Kliemann, The Dynamics of Control Birkhauser, 2000. doi: 10.1007/978-1-4612-1350-5. Google Scholar

[6]

A. Da Silva and C. Kawan, Hyperbolic chain control sets on flag manifolds, Journal of Dynamics and Control Systems, 22 (2016), 725-745. doi: 10.1007/s10883-015-9308-1. Google Scholar

[7]

A. Da Silva and C. Kawan, Invariance entropy of hyperbolic control sets, Discrete and Continuous Dynamical Systems, 36 (2016), 97-136. doi: 10.3934/dcds.2016.36.97. Google Scholar

[8]

V. Jurjevic, Geometric Control Theory Cambridge University Press, 1997.Google Scholar

[9]

V. Jurjevic and H. Sussmann, Control systems on Lie groups, Journal of Differential Equations, 12 (1972), 313-329. Google Scholar

[10]

C. Kawan, Invariance Entropy for Deterministic Control SystemsAn Introduction, Lecture Notes in Mathematics, 2089, Springer-Verlag, Berlin, 2013.Google Scholar

[11]

M. Patrao and L.A.B. San Martin, Semiflows on topological spaces: Chain transitivity and semigroups, Journal of Dynamics and Differential Equations, 19 (2007), 155-180. Google Scholar

[12]

L. A. B. San Martin, Order and domains of attractions of control sets in flag manifolds, Journal of Lie theory, 8 (1998), 335-350. Google Scholar

[13]

L. A. B. San Martin and L. Seco, Morse and Lyapunov spectra and dynamics on flag bundles, Ergod. Th. & Dynam. Sys., 30 (2010), 893-922. doi: 10.1017/S0143385709000285. Google Scholar

[14]

L. A. B. San Martin and P. A. Tonelli, Semigroup actions on Homogeneous Spaces, Semigroup Forum, 50 (1995), 59-88. doi: 10.1007/BF02573505. Google Scholar

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