Controllability of invariant control systems at uniform time

Víctor Ayala; José Ayala-Hoffmann; Ivan de Azevedo Tribuzy

Kybernetika (2009)

  • Volume: 45, Issue: 3, page 405-416
  • ISSN: 0023-5954

Abstract

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Let G be a compact and connected semisimple Lie group and Σ an invariant control systems on G . Our aim in this work is to give a new proof of Theorem 1 proved by Jurdjevic and Sussmann in [6]. Precisely, to find a positive time s Σ such that the system turns out controllable at uniform time s Σ . Our proof is different, elementary and the main argument comes directly from the definition of semisimple Lie group. The uniform time is not arbitrary. Finally, if A = ⋂ t > 0 A ( t , e ) denotes the reachable set from arbitrary uniform time, we conjecture that it is possible to determine A as the intersection of the isotropy groups of orbits of G -representations which contains exp ( 𝔷 ) , where 𝔷 is the Lie algebra determined by the control vectors.

How to cite

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Ayala, Víctor, Ayala-Hoffmann, José, and Azevedo Tribuzy, Ivan de. "Controllability of invariant control systems at uniform time." Kybernetika 45.3 (2009): 405-416. <http://eudml.org/doc/37674>.

@article{Ayala2009,
abstract = {Let $G$ be a compact and connected semisimple Lie group and $\Sigma $ an invariant control systems on $G$. Our aim in this work is to give a new proof of Theorem 1 proved by Jurdjevic and Sussmann in [6]. Precisely, to find a positive time $s_\{\Sigma \}$ such that the system turns out controllable at uniform time $s_\{\Sigma \}$. Our proof is different, elementary and the main argument comes directly from the definition of semisimple Lie group. The uniform time is not arbitrary. Finally, if $ A=\bigcap _\{ t > 0\}A(t,e)$ denotes the reachable set from arbitrary uniform time, we conjecture that it is possible to determine $A$ as the intersection of the isotropy groups of orbits of $G$-representations which contains $\exp (\mathfrak \{z\})$, where $\mathfrak \{z\}$ is the Lie algebra determined by the control vectors.},
author = {Ayala, Víctor, Ayala-Hoffmann, José, Azevedo Tribuzy, Ivan de},
journal = {Kybernetika},
keywords = {uniform-time; compact; semisimple; reverse-system; uniform-time; compact; semisimple; reverse-system},
language = {eng},
number = {3},
pages = {405-416},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Controllability of invariant control systems at uniform time},
url = {http://eudml.org/doc/37674},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Ayala, Víctor
AU - Ayala-Hoffmann, José
AU - Azevedo Tribuzy, Ivan de
TI - Controllability of invariant control systems at uniform time
JO - Kybernetika
PY - 2009
PB - Institute of Information Theory and Automation AS CR
VL - 45
IS - 3
SP - 405
EP - 416
AB - Let $G$ be a compact and connected semisimple Lie group and $\Sigma $ an invariant control systems on $G$. Our aim in this work is to give a new proof of Theorem 1 proved by Jurdjevic and Sussmann in [6]. Precisely, to find a positive time $s_{\Sigma }$ such that the system turns out controllable at uniform time $s_{\Sigma }$. Our proof is different, elementary and the main argument comes directly from the definition of semisimple Lie group. The uniform time is not arbitrary. Finally, if $ A=\bigcap _{ t > 0}A(t,e)$ denotes the reachable set from arbitrary uniform time, we conjecture that it is possible to determine $A$ as the intersection of the isotropy groups of orbits of $G$-representations which contains $\exp (\mathfrak {z})$, where $\mathfrak {z}$ is the Lie algebra determined by the control vectors.
LA - eng
KW - uniform-time; compact; semisimple; reverse-system; uniform-time; compact; semisimple; reverse-system
UR - http://eudml.org/doc/37674
ER -

References

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  2. Linear control systems on Lie groups and controllability, Amer. Math. Soc. Symposia in Pure Mathematics 64 (1999), 47–64. MR1654529
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  5. Controllability of nonlinear systems, J. Differential Equations 12 (1972), 95–116. MR0338882
  6. Control systems on Lie groups, J. Differential Equations 12 (1972), 313–329. MR0331185
  7. Support of diffusion processes and controllability problems, In: Proc. Internat. Symposium on Stochastic Differential Equations (K. Ito, ed.), Wiley, New York 1978, pp. 163–185. MR0536011
  8. Control Theory on Lie Groups, Lecture Notes SISSA, 2006. 
  9. Algebras de Lie, Editorial UNICAMP, Campinas, SP, 1999. 
  10. Uniform controllable sets of left-invariant vector fields on compact Lie groups, Systems Control Lett. 7 (1986), 213–216. Zbl0598.93005MR0847893
  11. Uniform controllable sets of left-invariant vector fields on non compact Lie groups, Systems Control Letters 6 (1986), 329–335. MR0821928
  12. Foundations of Differential Manifolds and Lie Groups, Scott Foreman, Glenview 1971. MR0295244

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