Geodesic
Control affine systems on solvable three-dimensional Lie groups, I
Archivum mathematicum, Tome 49 (2013) no. 3, pp. 187-197 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We seek to classify the full-rank left-invariant control affine systems evolving on solvable three-dimensional Lie groups. In this paper we consider only the cases corresponding to the solvable Lie algebras of types II, IV, and V in the Bianchi-Behr classification.
We seek to classify the full-rank left-invariant control affine systems evolving on solvable three-dimensional Lie groups. In this paper we consider only the cases corresponding to the solvable Lie algebras of types II, IV, and V in the Bianchi-Behr classification.
DOI : 10.5817/AM2013-3-187
Classification : 17B30, 93A10, 93B27
Keywords: left-invariant control system; (detached) feedback equivalence; affine subspace; solvable Lie algebra 
@article{10_5817_AM2013_3_187,
     author = {Biggs, Rory and Remsing, Claudiu C.},
     title = {Control affine systems on solvable three-dimensional {Lie} groups, {I}},
     journal = {Archivum mathematicum},
     pages = {187--197},
     year = {2013},
     volume = {49},
     number = {3},
     doi = {10.5817/AM2013-3-187},
     mrnumber = {3144181},
     zbl = {06321157},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2013-3-187/}
}
TY  - JOUR
AU  - Biggs, Rory
AU  - Remsing, Claudiu C.
TI  - Control affine systems on solvable three-dimensional Lie groups, I
JO  - Archivum mathematicum
PY  - 2013
SP  - 187
EP  - 197
VL  - 49
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.5817/AM2013-3-187/
DO  - 10.5817/AM2013-3-187
LA  - en
ID  - 10_5817_AM2013_3_187
ER  - 
%0 Journal Article
%A Biggs, Rory
%A Remsing, Claudiu C.
%T Control affine systems on solvable three-dimensional Lie groups, I
%J Archivum mathematicum
%D 2013
%P 187-197
%V 49
%N 3
%U http://geodesic.mathdoc.fr/articles/10.5817/AM2013-3-187/
%R 10.5817/AM2013-3-187
%G en
%F 10_5817_AM2013_3_187
Biggs, Rory; Remsing, Claudiu C. Control affine systems on solvable three-dimensional Lie groups, I. Archivum mathematicum, Tome 49 (2013) no. 3, pp. 187-197. doi: 10.5817/AM2013-3-187

[1] Agrachev, A. A., Sachkov, Y. L.: Control Theory from the Geometric Viewpoint. Springer Verlag, 2004. | MR | Zbl

[2] Biggs, R., Remsing, C. C.: On the equivalence of control systems on Lie groups. submitted.

[3] Biggs, R., Remsing, C. C.: A category of control systems. An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 20 (1) (2012), 355–368. | MR | Zbl

[4] Biggs, R., Remsing, C. C.: A note on the affine subspaces of three–dimensional Lie algebras. Bul. Acad. Ştiinţe Repub. Mold. Mat. no. 3 (2012), 45–52. | MR

[5] Biggs, R., Remsing, C. C.: Control affine systems on semisimple three–dimensional Lie groups. An. Ştiinţ. Univ. Al. I. Cuza Iaşi Mat. (N.S.) 59 (2) (2013), 399–414.

[6] Biggs, R., Remsing, C. C.: Control affine systems on solvable three–dimensional Lie groups, II. to appear in Note Mat. 33 (2013). | MR | Zbl

[7] Ha, K. Y., Lee, J. B.: Left invariant metrics and curvatures on simply connected three–dimensional Lie groups. Math. Nachr. 282 (6) (2009), 868–898. | DOI | MR | Zbl

[8] Harvey, A.: Automorphisms of the Bianchi model Lie groups. J. Math. Phys. 20 (2) (1979), 251–253. | DOI | MR

[9] Jakubczyk, B.: Equivalence and Invariants of Nonlinear Control Systems. Nonlinear Controllability and Optimal Control (Sussmann, H. J., ed.), M. Dekker, 1990, pp. 177–218. | MR | Zbl

[10] Jurdjevic, V.: Geometric Control Theory. Cambridge University Press, 1977. | MR

[11] Jurdjevic, V., Sussmann, H. J.: Control systems on Lie groups. J. Differential Equations 12 (1972), 313–329. | DOI | MR | Zbl

[12] Knapp, A. W.: Lie Groups beyond an Introduction. Progress in Mathematics, Birkhäuser, second ed., 2004. | MR

[13] Krasinski, A., et al., : The Bianchi classification in the Schücking–Behr approach. Gen. Relativity Gravitation 35 (3) (2003), 475–489. | DOI | MR | Zbl

[14] MacCallum, M. A. H.: On the Classification of the Real Four–Dimensional Lie Algebras. On Einstein's Path: Essays in Honour of E. Schücking (Harvey, A., ed.), Springer Verlag, 1999, pp. 299–317. | MR | Zbl

[15] Popovych, R. O., Boyco, V. M., Nesterenko, M. O., Lutfullin, M. W.: Realizations of real low–dimensional Lie algebras. J. Phys. A: Math. Gen. 36 (2003), 7337–7360. | DOI | MR

[16] Remsing, C. C.: Optimal control and Hamilton–Poisson formalism. Int. J. Pure Appl. Math. 59 (1) (2001), 11–17. | MR

Cité par Sources :