Control systems on semi-simple Lie groups and their homogeneous spaces

Velimir Jurdjevic; Ivan Kupka

Annales de l'institut Fourier (1981)

  • Volume: 31, Issue: 4, page 151-179
  • ISSN: 0373-0956

Abstract

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In the present paper, we consider the class of control systems which are induced by the action of a semi-simple Lie group on a manifold, and we give a sufficient condition which insures that such a system can be steered from any initial state to any final state by an admissible control. The class of systems considered contains, in particular, essentially all the bilinear systems. Our condition is semi-algebraic but unlike the celebrated Kalman criterion for linear systems, it is not necessary. In fact, it appears that there is no semi-algebraic necessary and sufficient condition in the bilinear case and that our criterion is in some sense optimal. This will be discussed in a future paper.

How to cite

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Jurdjevic, Velimir, and Kupka, Ivan. "Control systems on semi-simple Lie groups and their homogeneous spaces." Annales de l'institut Fourier 31.4 (1981): 151-179. <http://eudml.org/doc/74513>.

@article{Jurdjevic1981,
abstract = {In the present paper, we consider the class of control systems which are induced by the action of a semi-simple Lie group on a manifold, and we give a sufficient condition which insures that such a system can be steered from any initial state to any final state by an admissible control. The class of systems considered contains, in particular, essentially all the bilinear systems. Our condition is semi-algebraic but unlike the celebrated Kalman criterion for linear systems, it is not necessary. In fact, it appears that there is no semi-algebraic necessary and sufficient condition in the bilinear case and that our criterion is in some sense optimal. This will be discussed in a future paper.},
author = {Jurdjevic, Velimir, Kupka, Ivan},
journal = {Annales de l'institut Fourier},
keywords = {semi-simple Lie group; bilinear systems; semi-algebraic condition},
language = {eng},
number = {4},
pages = {151-179},
publisher = {Association des Annales de l'Institut Fourier},
title = {Control systems on semi-simple Lie groups and their homogeneous spaces},
url = {http://eudml.org/doc/74513},
volume = {31},
year = {1981},
}

TY - JOUR
AU - Jurdjevic, Velimir
AU - Kupka, Ivan
TI - Control systems on semi-simple Lie groups and their homogeneous spaces
JO - Annales de l'institut Fourier
PY - 1981
PB - Association des Annales de l'Institut Fourier
VL - 31
IS - 4
SP - 151
EP - 179
AB - In the present paper, we consider the class of control systems which are induced by the action of a semi-simple Lie group on a manifold, and we give a sufficient condition which insures that such a system can be steered from any initial state to any final state by an admissible control. The class of systems considered contains, in particular, essentially all the bilinear systems. Our condition is semi-algebraic but unlike the celebrated Kalman criterion for linear systems, it is not necessary. In fact, it appears that there is no semi-algebraic necessary and sufficient condition in the bilinear case and that our criterion is in some sense optimal. This will be discussed in a future paper.
LA - eng
KW - semi-simple Lie group; bilinear systems; semi-algebraic condition
UR - http://eudml.org/doc/74513
ER -

References

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  1. [1] N. BOURBAKI, Algèbre de Lie, Chap. VII-VIII, Hermann. 
  2. [2] A. BOREL-G. MOSTOW, On semi-simple automorphisms of Lie algebras, Ann. of Math., vol. 61 (1955), 389-405. Zbl0066.02401MR16,897d
  3. [3] J. DIXMIER, Enveloping algebras, North-Holland. Zbl0867.17001
  4. [4] H. FREUDENTHAL, Linear Lie groups, Academic Press. Zbl0377.22001
  5. [5] V. JURDJEVIC and I. KUPKA, Control systems subordinated to a group action : Accessibility, Journal of Diff. Equations, 39, 2 (1981), 186-211. Zbl0531.93008MR82f:93009
  6. [6] V. JURDJEVIC and H. SUSSMANN, Control systems on Lie groups, Journal of Diff. Equations, (12) (1972), 313-329. Zbl0237.93027MR48 #9519
  7. [7] A. KRENER, A generalization of Chow's theorem and the bang-bang theorem to non-linear control systems, SIAM J. Control, 11 (1973), 670-676. Zbl0243.93009
  8. [8] C. LOBRY, Contrôlabilité des systèmes non-linéaires, SIAM Journal on Control, 8 (1970), 573-605. Zbl0207.15201MR42 #6860
  9. [9] C. LOBRY, Contrôlabilité des systèmes non-linéaires, Proceedings of űOutils et modèles mathématiques pour l'automatique et l'analyse de systèmesƇ, C.N.R.S., Mai 1980, Centre Paul Langevin (CAES-CNRS), Aussois, France. 
  10. [10] B. LEVITT and H. SUSSMANN, On controllability by means of two vector fields, SIAM Journal on Control, 13 (1975), 1271-1281. Zbl0313.93006MR53 #6626
  11. [11] G. MOSTOW, Lie algebras and Lie groups, Mem. Amer. Math. Soc., n° 14. Zbl0080.25201
  12. [12] H. SUSSMANN and V. JURDJEVIC, Controllability of non-linear systems, Journal of Diff. Equations, 12 (1972), 95-116. Zbl0242.49040MR49 #3646

Citations in EuDML Documents

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  1. Mohamed Ghraiba, Semi-groupes intégraux de S L ( 2 , ℝ ) . Application à la théorie du contrôle
  2. Philippe Adda, Contrôlabilité des systèmes bilinéaires dans le plan
  3. Victor Bravo, Controllability of nilpotent systems
  4. V. Jurdjevic, Casimir elements and optimal control
  5. Mario Sigalotti, Jean-Claude Vivalda, Controllability properties of a class of systems modeling swimming microscopic organisms
  6. R. El Assoudi, J. Gauthier, I. Kupka, Controllability of right invariant systems on semi-simple Lie groups
  7. R. El Assoudi, J. P. Gauthier, I. A. K. Kupka, On subsemigroups of semisimple Lie groups
  8. Víctor Ayala, Eyüp Kizil, The covering semigroup of invariant control systems on Lie groups
  9. Ugo Boscain, Grégoire Charlot, Resonance of minimizers for n-level quantum systems with an arbitrary cost
  10. Ugo Boscain, Grégoire Charlot, Resonance of minimizers for -level quantum systems with an arbitrary cost

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