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uniform-time; compact; semisimple; reverse-system
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.
[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
[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
[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
[4] S. Helgason: Differential Geometry, Lie groups and Symmetric Spaces. Academic Press, New York 1978. MR 0514561 | Zbl 0993.53002
[5] V. Jurdjevic and H. J. Sussmann: Controllability of nonlinear systems. J. Differential Equations 12 (1972), 95–116. MR 0338882
[6] V. Jurdjevic and H. J. Sussmann: Control systems on Lie groups. J. Differential Equations 12 (1972), 313–329. MR 0331185
[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
[8] Y. Sachkov: Control Theory on Lie Groups. Lecture Notes SISSA, 2006.
[9] L. San Martin: Algebras de Lie. Editorial UNICAMP, Campinas, SP, 1999.
[10] F. Silva Leite: Uniform controllable sets of left-invariant vector fields on compact Lie groups. Systems Control Lett. 7 (1986), 213–216. MR 0847893 | Zbl 0598.93005
[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
[12] F. W. Warner: Foundations of Differential Manifolds and Lie Groups. Scott Foreman, Glenview 1971. MR 0295244
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