An estimation of the controllability time for single-input systems on compact Lie groups
ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 3, pp. 409-441.

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.

DOI : 10.1051/cocv:2006007
Classification : 22E46, 93B03
Mots clés : control systems, semi-simple Lie groups, riemannian geometry
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Agrachev, Andrei; Chambrion, Thomas. An estimation of the controllability time for single-input systems on compact Lie groups. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 3, pp. 409-441. doi : 10.1051/cocv:2006007. http://archive.numdam.org/articles/10.1051/cocv:2006007/

[1] J.F. Adams, Lectures on Lie groups. W.A. Benjamin, Inc., New York-Amsterdam (1969). | MR | Zbl

[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). | Zbl

[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. | MR | Zbl

[4] A.O. Barut and R. Raczka, Theory of group representations and applications. World Scientific Publishing Co., Singapore, second edn. (1986). | MR | Zbl

[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) 75 Astérisque, Soc. Math. France, Paris (1980) 19-45. | Numdam | Zbl

[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. | Zbl

[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. | Numdam | Zbl

[8] B. Bonnard, Contrôlabilité de systèmes mécaniques sur les groupes de Lie. SIAM J. Control Optim. 22 (1984) 711-722. | Zbl

[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. | Zbl

[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. | Zbl

[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. | Zbl

[12] U. Boscain and G. Charlot, Resonance of minimizers for n-level quantum systems with an arbitrary cost. ESAIM: COCV 10 (2004) 593-614. | Numdam | Zbl

[13] U. Boscain and Y. Chitour, On the minimum time problem for driftless left-invariant control systems on SO (3). Commun. Pure Appl. Anal. 1 (2002) 285-312. | Zbl

[14] R. Brockett, New issues in the mathematics of control, in Mathematics unlimited - 2001 and beyond. Springer, Berlin (2001), pp. 189-219. | Zbl

[15] D. D'Allessandro and M. Dahleh, Optimal control of two-level quantum systems. IEEE Trans. Automat. Control 46 (2001) 866-876. | Zbl

[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. | MR | Zbl

[17] R. El Assoudi and J.-P. Gauthier, Controllability of right invariant systems on real simple Lie groups of type F 4 ,G 2 ,C n , and B n . Math. Control Signals Syst. 1 (1988) 293-301. | Zbl

[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. | Zbl

[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. | Zbl

[20] R. El Assoudi, J.P. Gauthier and I.A.K. Kupka, On subsemigroups of semisimple Lie groups. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996) 117-133. | Numdam | Zbl

[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. | Zbl

[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). | Zbl

[24] S. Helgason, Differential geometry, Lie groups, and symmetric spaces 80, Pure Appl. Math., Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1978). | MR | Zbl

[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. | Zbl

[26] V. Jurdjevic, Optimal control, geometry, and mechanics, in Mathematical control theory. Springer, New York (1999) 227-267. | Zbl

[27] V. Jurdjevic and I. Kupka, Control systems on semisimple Lie groups and their homogeneous spaces. Ann. Inst. Fourier (Grenoble) 31 (1981) 151-179. | Numdam | Zbl

[28] V. Jurdjevic and I. Kupka, Control systems subordinated to a group action: accessibility. J. Differ. Equ. 39 (1981) 186-211. | Zbl

[29] V. Jurdjevic, Geometric control theory, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge 52 (1997). | MR | Zbl

[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. | Zbl

[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. | MR | Zbl

[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. A 65 (2002) 032301, 11. | MR

[33] 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. | Zbl

[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). | MR | Zbl

[35] J. Milnor, Curvatures of left invariant metrics on Lie groups. Advances Math. 21 (1976) 293-329. | Zbl

[36] T. Püttmann, Injectivity radius and diameter of the manifolds of flags in the projective planes. Math. Z. 246 (2004) 795-809. | Zbl

[37] Y.L. Sachkov, Controllability of invariant systems on Lie groups and homogeneous spaces. J. Math. Sci. 100 (2000) 2355-2427 Dynamical systems, 8. | Zbl

[38] H.J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems. J. Differ. Equ. 12 (1972) 95-116. | Zbl

[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. | MR | Zbl

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