Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T20:56:59.105Z Has data issue: false hasContentIssue false
  • English
  • Français

An estimation of the controllability time forsingle-input systems on compact Lie Groups

Published online by Cambridge University Press:  20 June 2006

Andrei Agrachev  and
Thomas Chambrion
Andrei Agrachev
Affiliation:
SISSA, Via Beirouth 2-4, 34013 Trieste, Italy; [email protected]
Thomas Chambrion
Affiliation:
SYSTeMS Group, University of Ghent, Technologiepark 914, 9052 Zwijnaarde, Belgium; [email protected]
Get access

Abstract

Geometric control theory and Riemannian techniques are used to describe the reachable set at time t of left invariant single-input control systems on semi-simple compact Lie groups and to estimate the minimal time needed to reach any point from identity. This method provides an effective way to give an upper and a lower bound for the minimal time needed to transfer a controlled quantum system with a drift from a given initial position to a given final position.The bounds include diameters of the flag manifolds; the latter arealso explicitly computed in the paper.

Keywords

Control systemssemi-simple Lie groupsRiemanniangeometry.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 3 , July 2006 , pp. 409 - 441
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J.F. Adams, Lectures on Lie groups. W.A. Benjamin, Inc., New York-Amsterdam (1969).
A.A. Agrachev, Introduction to optimal control theory, in Mathematical control theory, Part 1, 2 (Trieste, 2001), ICTP Lect. Notes, VIII, Abdus Salam Int. Cent. Theoret. Phys., Trieste (2002) 453–513 (electronic).
A.A. Agrachev and Y.L. Sachkov, Control theory from the geometric viewpoint, Encyclopaedia of Mathematical Sciences. 87 Springer-Verlag, Berlin (2004). Control Theory and Optimization, II.
A.O. Barut and R. Raczka, Theory of group representations and applications. World Scientific Publishing Co., Singapore, second edn. (1986).
B. Bonnard, V. Jurdjevic, I. Kupka and G. Sallet, Systèmes de champs de vecteurs transitifs sur les groupes de Lie semi-simples et leurs espaces homogènes, in Systems analysis (Conf., Bordeaux, 1978) 75 Astérisque, Soc. Math. France, Paris (1980) 19–45.
Bonnard, B., Jurdjevic, V., Kupka, I. and Sallet, G., Transitivity of families of invariant vector fields on the semidirect products of Lie groups. Trans. Amer. Math. Soc. 271 (1982) 525–535. CrossRef
B. Bonnard, Couples de générateurs de certaines sous-algèbres de Lie de l'algèbre de Lie symplectique affine, et applications. Publ. Dép. Math. (Lyon) 15 (1978) 1–36.
Bonnard, B., Contrôlabilité de systèmes mécaniques sur les groupes de Lie. SIAM J. Control Optim. 22 (1984) 711–722. CrossRef
Boscain, U., Chambrion, T. and Gauthier, J.-P., On the K + P problem for a three-level quantum system: optimality implies resonance. J. Dynam. Control Syst. 8 (2002) 547–572. CrossRef
U. Boscain, G. Charlot and J.-P. Gauthier, Optimal control of the Schrödinger equation with two or three levels, in Nonlinear and adaptive control (Sheffield 2001), Springer, Berlin, Lect. Not. Control Inform. Sci. 281 (2003) 33–43.
Boscain, U., Charlot, G., Gauthier, J.-P., Guérin, S. and Jauslin, H.-R., Optimal control in laser-induced population transfer for two- and three-level quantum systems. J. Math. Phys. 43 (2002) 2107–2132. CrossRef
Boscain, U. and Charlot, G., Resonance of minimizers for n-level quantum systems with an arbitrary cost. ESAIM: COCV 10 (2004) 593–614. CrossRef
Boscain, U. and Chitour, Y., On the minimum time problem for driftless left-invariant control systems on SO(2). Commun. Pure Appl. Anal. 1 (2002) 285–312.
R. Brockett, New issues in the mathematics of control, in Mathematics unlimited — 2001 and beyond. Springer, Berlin (2001), pp. 189–219.
D'Allessandro, D. and Dahleh, M., Optimal control of two-level quantum systems. IEEE Trans. Automat. Control 46 (2001) 866–876. CrossRef
M.P. do Carmo, Riemannian geometry, Mathematics: Theory & Applications. Birkhäuser Boston Inc., Boston, MA (1992). Translated from the second Portuguese edition by Francis Flaherty.
R. El Assoudi and J.-P. Gauthier, Controllability of right invariant systems on real simple Lie groups of type F4, G2, Cn, and Bn. Math. Control Signals Syst. 1 (1988) 293–301.
R. El Assoudi and J.-P. Gauthier, Controllability of right-invariant systems on semi-simple Lie groups, in New trends in nonlinear control theory (Nantes, 1988). Springer, Berlin, Lect. Notes Control Inform. Sci. 122 (1989) 54–64.
R. El Assoudi, J.P. Gauthier and I.A.K. Kupka, Controllability of right invariant systems on semi-simple Lie groups, in Geometry in nonlinear control and differential inclusions (Warsaw, 1993). Banach Center Publ., Polish Acad. Sci., Warsaw 32 (1995) 199–208.
El Assoudi, R., Gauthier, J.P. and Kupka, I.A.K., On subsemigroups of semisimple Lie groups. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996) 117–133. CrossRef
El Assoudi, R. and Gauthier, J.-P., Contrôlabilité sur l'espace quotient d'un groupe de Lie par un sous-groupe compact. C. R. Acad. Sci. Paris Sér. I Math. 311 (1990) 189–191.
A.L. Fradkov and A.N Churilov, Eds. Proceedings of the conference “Physics and Control” 2003 IEEE. August (2003).
J.-P. Gauthier, I. Kupka and G. Sallet, Controllability of right invariant systems on real simple Lie groups. Syst. Contr. Lett. 5 187–190 (1984).
S. Helgason, Differential geometry, Lie groups, and symmetric spaces 80, Pure Appl. Math., Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1978).
V. Jurdjevic, Optimal control problems on Lie groups: crossroads between geometry and mechanics, in Geometry of feedback and optimal control. Dekker, New York, Monogr. Textbooks Pure Appl. Math. 207 (1998) 257–303.
V. Jurdjevic, Optimal control, geometry, and mechanics, in Mathematical control theory. Springer, New York (1999) 227–267.
Jurdjevic, V. and Kupka, I., Control systems on semisimple Lie groups and their homogeneous spaces. Ann. Inst. Fourier (Grenoble) 31 (1981) 151–179. CrossRef
Jurdjevic, V. and Kupka, I., Control systems subordinated to a group action: accessibility. J. Differ. Equ. 39 (1981) 186–211. CrossRef
V. Jurdjevic, Geometric control theory, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge 52 (1997).
V. Jurdjevic, Lie determined systems and optimal problems with symmetries, in Geometric control and non-holonomic mechanics (Mexico City, 1996), Providence, RI. CMS Conf. Proc., Amer. Math. Soc. 25 (1998) 1–28.
A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications. 54 Cambridge University Press, Cambridge (1995). With a supplementary chapter by Katok and Leonardo Mendoza.
Khaneja, N., Glaser, S.J. and Brockett, R., Sub-Riemannian geometry and time optimal control of three spin systems: quantum gates and coherence transfer. Phys. Rev. A 65 (2002) 032301, 11. CrossRef
I. Kupka, Applications of semigroups to geometric control theory, in The analytical and topological theory of semigroups de Gruyter Exp. Math. de Gruyter, Berlin 1 (1990) 337–345.
J. Milnor, Morse theory. Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51. Princeton University Press, Princeton, N.J. (1963).
Milnor, J., Curvatures of left invariant metrics on Lie groups. Advances Math. 21 (1976) 293–329. CrossRef
Püttmann, T., Injectivity radius and diameter of the manifolds of flags in the projective planes. Math. Z. 246 (2004) 795–809. CrossRef
Sachkov, Y.L., Controllability of invariant systems on Lie groups and homogeneous spaces. J. Math. Sci. 100 (2000) 2355–2427 Dynamical systems, 8. CrossRef
Sussmann, H.J. and Jurdjevic, V., Controllability of nonlinear systems. J. Differ. Equ. 12 (1972) 95–116. CrossRef
V.S. Varadarajan, Lie groups, Lie algebras, and their representations. Prentice-Hall Inc., Englewood Cliffs, N.J. (1974). Prentice-Hall Series in Modern Analysis.