| Title:
|
Controllability of invariant control systems at uniform time (English) |
| Author:
|
Ayala, Víctor |
| Author:
|
Ayala-Hoffmann, José |
| Author:
|
Azevedo Tribuzy, Ivan de |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
45 |
| Issue:
|
3 |
| Year:
|
2009 |
| Pages:
|
405-416 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
uniform-time |
| Keyword:
|
compact |
| Keyword:
|
semisimple |
| Keyword:
|
reverse-system |
| MSC:
|
22E15 |
| MSC:
|
93B05 |
| MSC:
|
93C25 |
| idZBL:
|
Zbl 1165.93301 |
| idMR:
|
MR2543130 |
| . |
| Date available:
|
2010-06-02T18:38:32Z |
| Last updated:
|
2012-06-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140016 |
| . |
| Reference:
|
[1] V. Ayala and L. San Martin: Controllability properties of a class of control systems on Lie groups.Lectures Notes in Control and Inform. Sci. 1 (2001), 258, 83–92. MR 1806128 |
| Reference:
|
[2] V. Ayala and J. Tirao: Linear control systems on Lie groups and controllability.Amer. Math. Soc. Symposia in Pure Mathematics 64 (1999), 47–64. MR 1654529 |
| Reference:
|
[3] Domenico D’Alessandro: Small time controllability of systems on compact Lie groups and spin angular momentum.J. Math. Phys. 42 (2001) 9, 4488–4496. MR 1852638 |
| Reference:
|
[4] S. Helgason: Differential Geometry, Lie groups and Symmetric Spaces.Academic Press, New York 1978. Zbl 0993.53002, MR 0514561 |
| Reference:
|
[5] V. Jurdjevic and H. J. Sussmann: Controllability of nonlinear systems.J. Differential Equations 12 (1972), 95–116. MR 0338882 |
| Reference:
|
[6] V. Jurdjevic and H. J. Sussmann: Control systems on Lie groups.J. Differential Equations 12 (1972), 313–329. MR 0331185 |
| Reference:
|
[7] H. Kunita: 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. MR 0536011 |
| Reference:
|
[8] Y. Sachkov: Control Theory on Lie Groups.Lecture Notes SISSA, 2006. |
| Reference:
|
[9] L. San Martin: Algebras de Lie.Editorial UNICAMP, Campinas, SP, 1999. |
| Reference:
|
[10] F. Silva Leite: Uniform controllable sets of left-invariant vector fields on compact Lie groups.Systems Control Lett. 7 (1986), 213–216. Zbl 0598.93005, MR 0847893 |
| Reference:
|
[11] F. Silva Leite: Uniform controllable sets of left-invariant vector fields on non compact Lie groups.Systems Control Letters 6 (1986), 329–335. MR 0821928 |
| Reference:
|
[12] F. W. Warner: Foundations of Differential Manifolds and Lie Groups.Scott Foreman, Glenview 1971. MR 0295244 |
| . |
