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.
@article{bwmeta1.element.bwnjournal-article-bcpv50z1p41bwm,
author = {Bensalem, Naceurdine and Pelletier, Fernand},
title = {Some geometrical properties of infinite-dimensional bilinear controlled systems},
journal = {Banach Center Publications},
volume = {50},
year = {1999},
pages = {41-59},
zbl = {0971.93020},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv50z1p41bwm}
}
Bensalem, Naceurdine; Pelletier, Fernand. Some geometrical properties of infinite-dimensional bilinear controlled systems. Banach Center Publications, Tome 50 (1999) pp. 41-59. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv50z1p41bwm/
[000] [AS] A. Agrachev, A. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 635-690. | Zbl 0866.58023
[001] [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'.
[002] [AP] M. Arnaudon, S. Paycha, Stochastic tools on Hilbert manifolds: interplay with geometry and physics, Comm. Math. Phys. 187 (1997), 243-260. | Zbl 0888.58004
[003] [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. | Zbl 0992.20034
[004] [Bel] A. Bellaiche, Propriétés extrémales des géodésiques, Astérisque 84-85 (1981), 83-130. | Zbl 0529.53035
[005] [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.
[006] [BP] N. Bensalem, F. Pelletier, Approximation des extrémales pour un système bilinéaire en dimension infinie et applications, Preprint LAMA, Université de Savoie, 1998.
[007] [Bo] N. Bourbaki, Groupes et algèbres de Lie, Chapitres 1, 2 et 3, Hermann, Paris, 1971-72.
[008] [Eb] P. Eberlein, Geometry of 2-step nilpotent groups with a left invariant metric, Ann. Sci. École Norm. Sup. (4) 27 (1994), 611-660. | Zbl 0820.53047
[009] [Ek] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353. | Zbl 0286.49015
[010] [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. | Zbl 0832.93008
[011] [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. | Zbl 0829.93006
[012] [Ge] Z. Ge, Horizontal path spaces and Carnot-Carathéodory metrics, Pacific J. Math. 161 (1993), 255-286. | Zbl 0797.49033
[013] [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. | Zbl 0864.53025
[014] [Gu] A. Guichardet, Intégration, analyse hilbertienne, Ellipses, Paris, 1989.
[015] [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.
[016] [Ka2] A. Kaplan, Riemannian nilmanifolds attached to Clifford modules, Geom. Dedicata 11 (1981), 127-136. | Zbl 0495.53046
[017] [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. | Zbl 0835.22019
[018] [La] S. Lang, Differential Manifolds, Addison-Wesley, Reading, 1972.
[019] [LS] W. Liu, H. J. Sussmann, Shortest paths for sub-Riemannian metrics on rank-two distributions, Mem. Amer. Math. Soc. 118 (1995), no. 564.
[020] [Mo] R. Montgomery, A survey of singular curves in sub-Riemannian geometry, J. Dynam. Control Systems 1 (1995), 49-90. | Zbl 0941.53021
[021] [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.
[022] [OP] P. Orro, F. Pelletier, Propriétés géométriques de quelques distributions régulières, Prépublication 97, LAMA, Université de Savoie, 1997.
[023] [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.
[024] [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. | Zbl 0836.53025
[025] [PS] A. Pressley, G. Segal, Loop Groups, The Clarendon Press, Oxford Univ. Press, Oxford, 1986; Russian transl.: Mir, Moscow, 1990. | Zbl 0618.22011
[026] [X] C. Z. Xu, Exact observability and exponential stability of infinite-dimensional bilinear systems, Math. Control Signals Systems 9 (1996), 73-93. | Zbl 0862.93007
[027] [Z] M. Zhitomirskii, Rigid and abnormal line subdistributions of 2-distributions, J. Dynam. Control Systems 1 (1995), 253-294. | Zbl 0970.53020