# American Institute of Mathematical Sciences

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 Systems – An 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 Systems – An 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
 [1] Getachew K. Befekadu, Eduardo L. Pasiliao. On the hierarchical optimal control of a chain of distributed systems. Journal of Dynamics & Games, 2015, 2 (2) : 187-199. doi: 10.3934/jdg.2015.2.187 [2] Roberta Fabbri, Sylvia Novo, Carmen Núñez, Rafael Obaya. Null controllable sets and reachable sets for nonautonomous linear control systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1069-1094. doi: 10.3934/dcdss.2016042 [3] Thiago Ferraiol, Mauro Patrão, Lucas Seco. Jordan decomposition and dynamics on flag manifolds. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 923-947. doi: 10.3934/dcds.2010.26.923 [4] Anthony M. Bloch, Rohit Gupta, Ilya V. Kolmanovsky. Neighboring extremal optimal control for mechanical systems on Riemannian manifolds. Journal of Geometric Mechanics, 2016, 8 (3) : 257-272. doi: 10.3934/jgm.2016007 [5] Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455 [6] Adriano Da Silva, Christoph Kawan. Invariance entropy of hyperbolic control sets. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 97-136. doi: 10.3934/dcds.2016.36.97 [7] Hiromichi Nakayama, Takeo Noda. Minimal sets and chain recurrent sets of projective flows induced from minimal flows on $3$-manifolds. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 629-638. doi: 10.3934/dcds.2005.12.629 [8] Byungik Kahng, Miguel Mendes. The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems. Conference Publications, 2013, 2013 (special) : 393-406. doi: 10.3934/proc.2013.2013.393 [9] Robert Baier, Matthias Gerdts, Ilaria Xausa. Approximation of reachable sets using optimal control algorithms. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 519-548. doi: 10.3934/naco.2013.3.519 [10] Dietmar Szolnoki. Set oriented methods for computing reachable sets and control sets. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 361-382. doi: 10.3934/dcdsb.2003.3.361 [11] Andrew D. Lewis. Linearisation of tautological control systems. Journal of Geometric Mechanics, 2016, 8 (1) : 99-138. doi: 10.3934/jgm.2016.8.99 [12] Ralf Banisch, Carsten Hartmann. A sparse Markov chain approximation of LQ-type stochastic control problems. Mathematical Control & Related Fields, 2016, 6 (3) : 363-389. doi: 10.3934/mcrf.2016007 [13] Hamid Norouzi Nav, Mohammad Reza Jahed Motlagh, Ahmad Makui. Modeling and analyzing the chaotic behavior in supply chain networks: a control theoretic approach. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1123-1141. doi: 10.3934/jimo.2018002 [14] Sorin Micu, Jaime H. Ortega, Lionel Rosier, Bing-Yu Zhang. Control and stabilization of a family of Boussinesq systems. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 273-313. doi: 10.3934/dcds.2009.24.273 [15] Atte Aalto, Jarmo Malinen. Compositions of passive boundary control systems. Mathematical Control & Related Fields, 2013, 3 (1) : 1-19. doi: 10.3934/mcrf.2013.3.1 [16] Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii, Qingwen Hu. Selective Pyragas control of Hamiltonian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2019-2034. doi: 10.3934/dcdss.2019130 [17] Norbert Koksch, Stefan Siegmund. Feedback control via inertial manifolds for nonautonomous evolution equations. Communications on Pure & Applied Analysis, 2011, 10 (3) : 917-936. doi: 10.3934/cpaa.2011.10.917 [18] Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447 [19] Fritz Colonius. Invariance entropy, quasi-stationary measures and control sets. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2093-2123. doi: 10.3934/dcds.2018086 [20] Tayel Dabbous. Adaptive control of nonlinear systems using fuzzy systems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 861-880. doi: 10.3934/jimo.2010.6.861

2018 Impact Factor: 1.143

## Metrics

• PDF downloads (10)
• HTML views (1)
• Cited by (1)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]