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

Annales de l'institut Fourier (1981)

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

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topJurdjevic, 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

top- [1] N. BOURBAKI, Algèbre de Lie, Chap. VII-VIII, Hermann.
- [2] A. BOREL-G. MOSTOW, On semi-simple automorphisms of Lie algebras, Ann. of Math., vol. 61 (1955), 389-405. Zbl0066.02401MR16,897d
- [3] J. DIXMIER, Enveloping algebras, North-Holland. Zbl0867.17001
- [4] H. FREUDENTHAL, Linear Lie groups, Academic Press. Zbl0377.22001
- [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] V. JURDJEVIC and H. SUSSMANN, Control systems on Lie groups, Journal of Diff. Equations, (12) (1972), 313-329. Zbl0237.93027MR48 #9519
- [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] C. LOBRY, Contrôlabilité des systèmes non-linéaires, SIAM Journal on Control, 8 (1970), 573-605. Zbl0207.15201MR42 #6860
- [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] 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] G. MOSTOW, Lie algebras and Lie groups, Mem. Amer. Math. Soc., n° 14. Zbl0080.25201
- [12] H. SUSSMANN and V. JURDJEVIC, Controllability of non-linear systems, Journal of Diff. Equations, 12 (1972), 95-116. Zbl0242.49040MR49 #3646

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