×

An estimation of the controllability time for single-input systems on compact Lie groups. (English) Zbl 1106.93006

Summary: 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.

MSC:

93B05 Controllability
22E15 General properties and structure of real Lie groups
49K15 Optimality conditions for problems involving ordinary differential equations
93B29 Differential-geometric methods in systems theory (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] J.F. Adams , Lectures on Lie groups . W.A. Benjamin, Inc., New York-Amsterdam ( 1969 ). MR 252560 | Zbl 0206.31604 · Zbl 0206.31604
[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 1034.93010 · Zbl 1034.93010
[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 2062547 | Zbl 1062.93001 · Zbl 1062.93001
[4] A.O. Barut and R. Raczka , Theory of group representations and applications . World Scientific Publishing Co., Singapore, second edn. ( 1986 ). MR 889252 | Zbl 0644.22011 · Zbl 0644.22011
[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 . Zbl 0458.93009 · Zbl 0458.93009
[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 0519.49023 · Zbl 0519.49023 · doi:10.2307/1998897
[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 . Zbl 0434.93013 · Zbl 0434.93013
[8] B. Bonnard , Contrôlabilité de systèmes mécaniques sur les groupes de Lie . SIAM J. Control Optim. 22 ( 1984 ) 711 - 722 . Zbl 0549.93009 · Zbl 0549.93009 · doi:10.1137/0322045
[9] U. Boscain , T. Chambrion and J.-P. Gauthier , On the \(\mathrm K+P\) problem for a three-level quantum system: optimality implies resonance . J. Dynam. Control Syst. 8 ( 2002 ) 547 - 572 . Zbl 1022.53028 · Zbl 1022.53028 · doi:10.1023/A:1020767419671
[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 1009.93060 · Zbl 1009.93060
[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 1059.81195 · Zbl 1059.81195 · doi:10.1063/1.1465516
[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 1072.49002 · Zbl 1072.49002 · doi:10.1051/cocv:2004022
[13] U. Boscain and Y. Chitour , On the minimum time problem for driftless left-invariant control systems on \({\mathrm SO}(3)\) . Commun. Pure Appl. Anal. 1 ( 2002 ) 285 - 312 . Zbl 1034.49015 · Zbl 1034.49015 · doi:10.3934/cpaa.2002.1.285
[14] R. Brockett , New issues in the mathematics of control , in Mathematics unlimited - 2001 and beyond. Springer, Berlin ( 2001 ), pp. 189 - 219 . Zbl 1005.93002 · Zbl 1005.93002
[15] D. D’Allessandro and M. Dahleh , Optimal control of two-level quantum systems . IEEE Trans. Automat. Control 46 ( 2001 ) 866 - 876 . Zbl 0993.81070 · Zbl 0993.81070 · doi:10.1109/9.928587
[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 1138207 | Zbl 0752.53001 · Zbl 0752.53001
[17] R. El Assoudi and J.-P. Gauthier , Controllability of right invariant systems on real simple Lie groups of type \(\mathrm F_ 4,\;G_ 2,\;C_ n,\) and \(\mathrm B_ n\) . Math. Control Signals Syst. 1 ( 1988 ) 293 - 301 . Zbl 0672.93009 · Zbl 0672.93009 · doi:10.1007/BF02551290
[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 0682.93009 · Zbl 0682.93009
[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 0839.93019 · Zbl 0839.93019
[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 0848.93006 · Zbl 0848.93006
[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 0717.93004 · Zbl 0717.93004
[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 0552.93010 · Zbl 0552.93010 · doi:10.1016/S0167-6911(84)80101-2
[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 514561 | Zbl 0451.53038 · Zbl 0451.53038
[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 0925.93138 · Zbl 0925.93138
[26] V. Jurdjevic , Optimal control, geometry, and mechanics , in Mathematical control theory. Springer, New York ( 1999 ) 227 - 267 . Zbl 1047.93506 · Zbl 1047.93506
[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 0453.93011 · Zbl 0453.93011 · doi:10.5802/aif.853
[28] V. Jurdjevic and I. Kupka , Control systems subordinated to a group action: accessibility . J. Differ. Equ. 39 ( 1981 ) 186 - 211 . Zbl 0531.93008 · Zbl 0531.93008 · doi:10.1016/0022-0396(81)90072-3
[29] V. Jurdjevic , Geometric control theory , Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge 52 ( 1997 ). MR 1425878 | Zbl 0940.93005 · Zbl 0940.93005
[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 1047.93507 · Zbl 1047.93507
[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 1326374 | Zbl 0878.58020 · Zbl 0878.58020
[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 1891763
[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 0715.93006 · Zbl 0715.93006
[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 163331 | Zbl 0108.10401 · Zbl 0108.10401
[35] J. Milnor , Curvatures of left invariant metrics on Lie groups . Advances Math. 21 ( 1976 ) 293 - 329 . Zbl 0341.53030 · Zbl 0341.53030 · doi:10.1016/S0001-8708(76)80002-3
[36] T. Püttmann , Injectivity radius and diameter of the manifolds of flags in the projective planes . Math. Z. 246 ( 2004 ) 795 - 809 . Zbl 1069.53034 · Zbl 1069.53034 · doi:10.1007/s00209-003-0613-0
[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 1073.93511 · Zbl 1073.93511
[38] H.J. Sussmann and V. Jurdjevic , Controllability of nonlinear systems . J. Differ. Equ. 12 ( 1972 ) 95 - 116 . Zbl 0242.49040 · Zbl 0242.49040 · doi:10.1016/0022-0396(72)90007-1
[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 376938 | Zbl 0371.22001 · Zbl 0371.22001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.