Geometric control theory and riemannian techniques are used to describe the reachable set at time 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.
Keywords: control systems, semi-simple Lie groups, riemannian geometry
@article{COCV_2006__12_3_409_0, author = {Agrachev, Andrei and Chambrion, Thomas}, title = {An estimation of the controllability time for single-input systems on compact {Lie} groups}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {409--441}, publisher = {EDP-Sciences}, volume = {12}, number = {3}, year = {2006}, doi = {10.1051/cocv:2006007}, mrnumber = {2224821}, zbl = {1106.93006}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2006007/} }
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 PY - 2006 SP - 409 EP - 441 VL - 12 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2006007/ DO - 10.1051/cocv:2006007 LA - en ID - COCV_2006__12_3_409_0 ER -
%0 Journal Article %A Agrachev, Andrei %A Chambrion, Thomas %T An estimation of the controllability time for single-input systems on compact Lie groups %J ESAIM: Control, Optimisation and Calculus of Variations %D 2006 %P 409-441 %V 12 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2006007/ %R 10.1051/cocv:2006007 %G en %F COCV_2006__12_3_409_0
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, Volume 12 (2006) no. 3, pp. 409-441. doi : 10.1051/cocv:2006007. http://www.numdam.org/articles/10.1051/cocv:2006007/
[1] Lectures on Lie groups. W.A. Benjamin, Inc., New York-Amsterdam (1969). | MR | Zbl
,[2] 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] Control theory from the geometric viewpoint, Encyclopaedia of Mathematical Sciences. 87 Springer-Verlag, Berlin (2004). Control Theory and Optimization, II. | MR | Zbl
and ,[4] Theory of group representations and applications. World Scientific Publishing Co., Singapore, second edn. (1986). | MR | Zbl
and ,[5] 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
, , and ,[6] Transitivity of families of invariant vector fields on the semidirect products of Lie groups. Trans. Amer. Math. Soc. 271 (1982) 525-535. | Zbl
, , and ,[7] 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] Contrôlabilité de systèmes mécaniques sur les groupes de Lie. SIAM J. Control Optim. 22 (1984) 711-722. | Zbl
,[9] On the problem for a three-level quantum system: optimality implies resonance. J. Dynam. Control Syst. 8 (2002) 547-572. | Zbl
, and ,[10] 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
, and ,[11] Optimal control in laser-induced population transfer for two- and three-level quantum systems. J. Math. Phys. 43 (2002) 2107-2132. | Zbl
, , , and ,[12] Resonance of minimizers for -level quantum systems with an arbitrary cost. ESAIM: COCV 10 (2004) 593-614. | Numdam | Zbl
and ,[13] On the minimum time problem for driftless left-invariant control systems on . Commun. Pure Appl. Anal. 1 (2002) 285-312. | Zbl
and ,[14] New issues in the mathematics of control, in Mathematics unlimited - 2001 and beyond. Springer, Berlin (2001), pp. 189-219. | Zbl
,[15] Optimal control of two-level quantum systems. IEEE Trans. Automat. Control 46 (2001) 866-876. | Zbl
and ,[16] Riemannian geometry, Mathematics: Theory & Applications. Birkhäuser Boston Inc., Boston, MA (1992). Translated from the second Portuguese edition by Francis Flaherty. | MR | Zbl
,[17] Controllability of right invariant systems on real simple Lie groups of type and . Math. Control Signals Syst. 1 (1988) 293-301. | Zbl
and ,[18] 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
and ,[19] 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
, and ,[20] On subsemigroups of semisimple Lie groups. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996) 117-133. | Numdam | Zbl
, and ,[21] 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
and ,[22]
and , Eds. Proceedings of the conference “Physics and Control” 2003 IEEE. August (2003).[23] Controllability of right invariant systems on real simple Lie groups. Syst. Contr. Lett. 5 187-190 (1984). | Zbl
, and ,[24] Differential geometry, Lie groups, and symmetric spaces 80, Pure Appl. Math., Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1978). | MR | Zbl
,[25] 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] Optimal control, geometry, and mechanics, in Mathematical control theory. Springer, New York (1999) 227-267. | Zbl
,[27] Control systems on semisimple Lie groups and their homogeneous spaces. Ann. Inst. Fourier (Grenoble) 31 (1981) 151-179. | Numdam | Zbl
and ,[28] Control systems subordinated to a group action: accessibility. J. Differ. Equ. 39 (1981) 186-211. | Zbl
and ,[29] Geometric control theory, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge 52 (1997). | MR | Zbl
,[30] 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] 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
and ,[32] Sub-Riemannian geometry and time optimal control of three spin systems: quantum gates and coherence transfer. Phys. Rev. A 65 (2002) 032301, 11. | MR
, and ,[33] 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] 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] Curvatures of left invariant metrics on Lie groups. Advances Math. 21 (1976) 293-329. | Zbl
,[36] Injectivity radius and diameter of the manifolds of flags in the projective planes. Math. Z. 246 (2004) 795-809. | Zbl
,[37] Controllability of invariant systems on Lie groups and homogeneous spaces. J. Math. Sci. 100 (2000) 2355-2427 Dynamical systems, 8. | Zbl
,[38] Controllability of nonlinear systems. J. Differ. Equ. 12 (1972) 95-116. | Zbl
and ,[39] 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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