Some geometrical properties of infinite-dimensional bilinear controlled systems

Naceurdine Bensalem; Fernand Pelletier

Banach Center Publications (1999)

  • Volume: 50, Issue: 1, page 41-59
  • ISSN: 0137-6934

Abstract

top
The study of controlled infinite-dimensional systems gives rise to many papers (see for instance [GXL], [GXB], [X]) but it is also motivated by various mathematical problems: partial differential equations ([BP]), sub-Riemannian geometry on infinite-dimensional manifolds ([Gr]), deformations in loop-spaces ([AP], [PS]). The first difference between finite and infinite-dimensional cases is that solutions in general do not exist (even locally) for every given control function. The aim of this paper is to study "infinite bilinear systems" on Hilbert spaces for which such a solution always exists. Moreover, to this particular class of controlled systems a nilpotent Lie algebra of degree 2 is naturally associated. On the other hand, given a Hilbertian nilpotent Lie algebra G of degree 2 we can associate to it in a natural way a bilinear system corresponding to left invariant distributions on a connected Lie groups G whose Lie algebra is G. The first result we obtain is an accessibility one which can be considered as a version of Chow's theorem in this situation. If we consider infinite-dimensional time optimal controlled systems the optimal trajectories are always abnormal curves which can be defined as in the finite-dimensional case. The second result of this paper is to give a "localization" of such curves: each of them is actually "normal" in some induced system on a submanifold. Finally we illustrate these results in the case of classical infinite generalized Heisenberg algebras.

How to cite

top

Bensalem, Naceurdine, and Pelletier, Fernand. "Some geometrical properties of infinite-dimensional bilinear controlled systems." Banach Center Publications 50.1 (1999): 41-59. <http://eudml.org/doc/209017>.

@article{Bensalem1999,
abstract = {The study of controlled infinite-dimensional systems gives rise to many papers (see for instance [GXL], [GXB], [X]) but it is also motivated by various mathematical problems: partial differential equations ([BP]), sub-Riemannian geometry on infinite-dimensional manifolds ([Gr]), deformations in loop-spaces ([AP], [PS]). The first difference between finite and infinite-dimensional cases is that solutions in general do not exist (even locally) for every given control function. The aim of this paper is to study "infinite bilinear systems" on Hilbert spaces for which such a solution always exists. Moreover, to this particular class of controlled systems a nilpotent Lie algebra of degree 2 is naturally associated. On the other hand, given a Hilbertian nilpotent Lie algebra G of degree 2 we can associate to it in a natural way a bilinear system corresponding to left invariant distributions on a connected Lie groups G whose Lie algebra is G. The first result we obtain is an accessibility one which can be considered as a version of Chow's theorem in this situation. If we consider infinite-dimensional time optimal controlled systems the optimal trajectories are always abnormal curves which can be defined as in the finite-dimensional case. The second result of this paper is to give a "localization" of such curves: each of them is actually "normal" in some induced system on a submanifold. Finally we illustrate these results in the case of classical infinite generalized Heisenberg algebras.},
author = {Bensalem, Naceurdine, Pelletier, Fernand},
journal = {Banach Center Publications},
keywords = {invariant control systems on infinite-dimensional step-two nilpotent Lie groups; local controllability},
language = {eng},
number = {1},
pages = {41-59},
title = {Some geometrical properties of infinite-dimensional bilinear controlled systems},
url = {http://eudml.org/doc/209017},
volume = {50},
year = {1999},
}

TY - JOUR
AU - Bensalem, Naceurdine
AU - Pelletier, Fernand
TI - Some geometrical properties of infinite-dimensional bilinear controlled systems
JO - Banach Center Publications
PY - 1999
VL - 50
IS - 1
SP - 41
EP - 59
AB - The study of controlled infinite-dimensional systems gives rise to many papers (see for instance [GXL], [GXB], [X]) but it is also motivated by various mathematical problems: partial differential equations ([BP]), sub-Riemannian geometry on infinite-dimensional manifolds ([Gr]), deformations in loop-spaces ([AP], [PS]). The first difference between finite and infinite-dimensional cases is that solutions in general do not exist (even locally) for every given control function. The aim of this paper is to study "infinite bilinear systems" on Hilbert spaces for which such a solution always exists. Moreover, to this particular class of controlled systems a nilpotent Lie algebra of degree 2 is naturally associated. On the other hand, given a Hilbertian nilpotent Lie algebra G of degree 2 we can associate to it in a natural way a bilinear system corresponding to left invariant distributions on a connected Lie groups G whose Lie algebra is G. The first result we obtain is an accessibility one which can be considered as a version of Chow's theorem in this situation. If we consider infinite-dimensional time optimal controlled systems the optimal trajectories are always abnormal curves which can be defined as in the finite-dimensional case. The second result of this paper is to give a "localization" of such curves: each of them is actually "normal" in some induced system on a submanifold. Finally we illustrate these results in the case of classical infinite generalized Heisenberg algebras.
LA - eng
KW - invariant control systems on infinite-dimensional step-two nilpotent Lie groups; local controllability
UR - http://eudml.org/doc/209017
ER -

References

top
  1. [AS] A. Agrachev, A. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 635-690. Zbl0866.58023
  2. [AOP] M. Alcheikh, P. Orro, F. Pelletier, Singularité de l'application extrémité pour les chemins horizontaux en géométrie sous-riemannienne, Prépublication 95-09a LAMA, Université de Savoie, 1995; to be published in: Actes du colloque 'Géométrie sous riemannienne et singularités'. 
  3. [AP] M. Arnaudon, S. Paycha, Stochastic tools on Hilbert manifolds: interplay with geometry and physics, Comm. Math. Phys. 187 (1997), 243-260. Zbl0888.58004
  4. [BG] V. Baranovsky, V. Ginzburg, Conjugacy classes in loop groups and G-bundles on elliptic curves, Internat. Math. Res. Notices 1996, no. 15, 733-751. Zbl0992.20034
  5. [Bel] A. Bellaiche, Propriétés extrémales des géodésiques, Astérisque 84-85 (1981), 83-130. Zbl0529.53035
  6. [Be] N. Bensalem, Localisation des courbes anormales et problème d'accessibilité sur un groupe de Lie hilbertien nilpotent de degré 2, Thèse de doctorat, Université de Savoie, 1998. 
  7. [BP] N. Bensalem, F. Pelletier, Approximation des extrémales pour un système bilinéaire en dimension infinie et applications, Preprint LAMA, Université de Savoie, 1998. 
  8. [Bo] N. Bourbaki, Groupes et algèbres de Lie, Chapitres 1, 2 et 3, Hermann, Paris, 1971-72. 
  9. [Eb] P. Eberlein, Geometry of 2-step nilpotent groups with a left invariant metric, Ann. Sci. École Norm. Sup. (4) 27 (1994), 611-660. Zbl0820.53047
  10. [Ek] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353. Zbl0286.49015
  11. [GXB] J. P. Gauthier, C. Z. Xu, A. Bounabat, An observer for infinite-dimensional skew-adjoint bilinear systems, J. Math. Systems Estim. Control 5 (1995), 119-122. Zbl0832.93008
  12. [GXL] J. P. Gauthier, C. Z. Xu, P. Ligarius, An observer for infinite-dimensional dissipative bilinear systems, Comput. Math. Appl. 29 no. 7 (1995), 13-21. Zbl0829.93006
  13. [Ge] Z. Ge, Horizontal path spaces and Carnot-Carathéodory metrics, Pacific J. Math. 161 (1993), 255-286. Zbl0797.49033
  14. [Gr] M. Gromov, Carnot-Carathéodory spaces seen from within, in: Sub-Riemannian Geometry, A. Bellaï che and J.-J. Risler (eds.), Progr. Math. 144, Birkhäuser, Boston, 1996, 79-323. Zbl0864.53025
  15. [Gu] A. Guichardet, Intégration, analyse hilbertienne, Ellipses, Paris, 1989. 
  16. [Ka1] A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), 147-153. 
  17. [Ka2] A. Kaplan, Riemannian nilmanifolds attached to Clifford modules, Geom. Dedicata 11 (1981), 127-136. Zbl0495.53046
  18. [Ko] H. Konno, Geometry of loop groups and Wess-Zumino-Witten models, in: Symplectic Geometry and Quantization, Y. Maeda et al. (eds.), Contemp. Math. 179, Amer. Math. Soc., Providence, 1994, 139-160. Zbl0835.22019
  19. [La] S. Lang, Differential Manifolds, Addison-Wesley, Reading, 1972. 
  20. [LS] W. Liu, H. J. Sussmann, Shortest paths for sub-Riemannian metrics on rank-two distributions, Mem. Amer. Math. Soc. 118 (1995), no. 564. 
  21. [Mo] R. Montgomery, A survey of singular curves in sub-Riemannian geometry, J. Dynam. Control Systems 1 (1995), 49-90. Zbl0941.53021
  22. [MP] P. Mormul, F. Pelletier, Contrôlabilité complète par des courbes anormales par morceaux d'une distribution de rang 3 générique sur des variétés connexes de dimension 5 et 6, Bull. Polish Acad. Sci. Math. 45 (1997), 399-418. 
  23. [OP] P. Orro, F. Pelletier, Propriétés géométriques de quelques distributions régulières, Prépublication 97, LAMA, Université de Savoie, 1997. 
  24. [PV1] F. Pelletier, L. Valère Bouche, Le problème des géodésiques en géométrie sous-riemannienne singulière, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 71-76. 
  25. [PV2] F. Pelletier, L. Valère Bouche, Abnormality of trajectory in sub-Riemannian structure, in: Geometry in Nonlinear Control and Differential Inclusions, Banach Center Publ. 32, Warsaw, 1995, 301-317. Zbl0836.53025
  26. [PS] A. Pressley, G. Segal, Loop Groups, The Clarendon Press, Oxford Univ. Press, Oxford, 1986; Russian transl.: Mir, Moscow, 1990. Zbl0618.22011
  27. [X] C. Z. Xu, Exact observability and exponential stability of infinite-dimensional bilinear systems, Math. Control Signals Systems 9 (1996), 73-93. Zbl0862.93007
  28. [Z] M. Zhitomirskii, Rigid and abnormal line subdistributions of 2-distributions, J. Dynam. Control Systems 1 (1995), 253-294. Zbl0970.53020

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.