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

Andrei Agrachev; Thomas Chambrion

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 3, page 409-441
  • ISSN: 1292-8119

Abstract

top
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 are also explicitly computed in the paper.

How to cite

top

Agrachev, Andrei, and Chambrion, Thomas. "An estimation of the controllability time for single-input systems on compact Lie Groups." ESAIM: Control, Optimisation and Calculus of Variations 12.3 (2006): 409-441. <http://eudml.org/doc/249680>.

@article{Agrachev2006,
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 are also explicitly computed in the paper. },
author = {Agrachev, Andrei, Chambrion, Thomas},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Control systems; semi-simple Lie groups; Riemannian geometry. ; Riemannian geometry},
language = {eng},
month = {6},
number = {3},
pages = {409-441},
publisher = {EDP Sciences},
title = {An estimation of the controllability time for single-input systems on compact Lie Groups},
url = {http://eudml.org/doc/249680},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Agrachev, Andrei
AU - Chambrion, Thomas
TI - An estimation of the controllability time for single-input systems on compact Lie Groups
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/6//
PB - EDP Sciences
VL - 12
IS - 3
SP - 409
EP - 441
AB - 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 are also explicitly computed in the paper.
LA - eng
KW - Control systems; semi-simple Lie groups; Riemannian geometry. ; Riemannian geometry
UR - http://eudml.org/doc/249680
ER -

References

top
  1. J.F. Adams, Lectures on Lie groups. W.A. Benjamin, Inc., New York-Amsterdam (1969).  
  2. 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).  
  3. 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.  
  4. A.O. Barut and R. Raczka, Theory of group representations and applications. World Scientific Publishing Co., Singapore, second edn. (1986).  
  5. 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)75Astérisque, Soc. Math. France, Paris (1980) 19–45.  
  6. B. Bonnard, V. Jurdjevic, I. Kupka and G. Sallet, Transitivity of families of invariant vector fields on the semidirect products of Lie groups. Trans. Amer. Math. Soc.271 (1982) 525–535.  
  7. 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.  
  8. B. Bonnard, Contrôlabilité de systèmes mécaniques sur les groupes de Lie. SIAM J. Control Optim.22 (1984) 711–722.  
  9. U. Boscain, T. Chambrion and J.-P. Gauthier, On the K + P problem for a three-level quantum system: optimality implies resonance. J. Dynam. Control Syst.8 (2002) 547–572.  
  10. 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.  
  11. U. Boscain, G. Charlot, J.-P. Gauthier, S. Guérin and H.-R. Jauslin, Optimal control in laser-induced population transfer for two- and three-level quantum systems. J. Math. Phys.43 (2002) 2107–2132.  
  12. U. Boscain and G. Charlot, Resonance of minimizers for n-level quantum systems with an arbitrary cost. ESAIM: COCV10 (2004) 593–614.  
  13. U. Boscain and Y. Chitour, On the minimum time problem for driftless left-invariant control systems on SO(2). Commun. Pure Appl. Anal.1 (2002) 285–312.  
  14. R. Brockett, New issues in the mathematics of control, in Mathematics unlimited — 2001 and beyond. Springer, Berlin (2001), pp. 189–219.  
  15. D. D'Allessandro and M. Dahleh, Optimal control of two-level quantum systems. IEEE Trans. Automat. Control46 (2001) 866–876.  
  16. M.P. do Carmo, Riemannian geometry, Mathematics: Theory & Applications. Birkhäuser Boston Inc., Boston, MA (1992). Translated from the second Portuguese edition by Francis Flaherty.  
  17. 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.  
  18. 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.  
  19. 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.  
  20. R. El Assoudi, J.P. Gauthier and I.A.K. Kupka, On subsemigroups of semisimple Lie groups. Ann. Inst. H. Poincaré Anal. Non Linéaire13 (1996) 117–133.  
  21. R. El Assoudi and J.-P. Gauthier, 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.  
  22. A.L. Fradkov and A.N Churilov, Eds. Proceedings of the conference “Physics and Control” 2003 IEEE. August (2003).  
  23. 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).  
  24. S. Helgason, Differential geometry, Lie groups, and symmetric spaces 80, Pure Appl. Math., Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1978).  
  25. 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.  
  26. V. Jurdjevic, Optimal control, geometry, and mechanics, in Mathematical control theory. Springer, New York (1999) 227–267.  
  27. V. Jurdjevic and I. Kupka, Control systems on semisimple Lie groups and their homogeneous spaces. Ann. Inst. Fourier (Grenoble)31 (1981) 151–179.  
  28. V. Jurdjevic and I. Kupka, Control systems subordinated to a group action: accessibility. J. Differ. Equ.39 (1981) 186–211.  
  29. V. Jurdjevic, Geometric control theory, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge 52 (1997).  
  30. 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.  
  31. 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.  
  32. N. Khaneja, S.J. Glaser and R. Brockett, Sub-Riemannian geometry and time optimal control of three spin systems: quantum gates and coherence transfer. Phys. Rev. A65 (2002) 032301, 11.  
  33. I. Kupka, Applications of semigroups to geometric control theory, in The analytical and topological theory of semigroupsde Gruyter Exp. Math. de Gruyter, Berlin 1 (1990) 337–345.  
  34. 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).  
  35. J. Milnor, Curvatures of left invariant metrics on Lie groups. Advances Math.21 (1976) 293–329.  
  36. T. Püttmann, Injectivity radius and diameter of the manifolds of flags in the projective planes. Math. Z.246 (2004) 795–809.  
  37. Y.L. Sachkov, Controllability of invariant systems on Lie groups and homogeneous spaces. J. Math. Sci.100 (2000) 2355–2427 Dynamical systems, 8.  
  38. H.J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems. J. Differ. Equ.12 (1972) 95–116.  
  39. V.S. Varadarajan, Lie groups, Lie algebras, and their representations. Prentice-Hall Inc., Englewood Cliffs, N.J. (1974). Prentice-Hall Series in Modern Analysis.  

NotesEmbed ?

top

You must be logged in to post comments.