Analysis of Lyapunov method for control of quantum states

IEEE Transactions on Automatic Control

Volumn 55, Issue 10, 2010, Pages 2259-2270

  Wang, Xiaoting ^a   Schirmer, Sophie G ^a  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

Eigenvalue analysis; LaSalle invariance principle; quantum systems; trajectory tracking

Indexed keywords

A-CENTER; ASYMPTOTICALLY STABLE; CONTROL DESIGN; CONTROL DYNAMIC; CRITICAL POINT ANALYSIS; CRITICAL POINTS; DENSITY OPERATORS; EIGENVALUE ANALYSIS; GLOBAL MAXIMUM; GLOBAL MINIMA; GRADIENT FLOW; HILBERT-SCHMIDT DISTANCE; INVARIANT SET; LASALLE INVARIANCE PRINCIPLE; LASALLE'S INVARIANCE PRINCIPLE; LINEARIZED DYNAMICS; LINEARIZED SYSTEMS; MORSE FUNCTIONS; NONSTATIONARY; QUANTUM STATE; QUANTUM SYSTEM; STABLE MANIFOLD; STATE SPACE; SUFFICIENT CONDITIONS; TARGET STATE; TRAJECTORY TRACKING; TRAJECTORY TRACKING PROBLEMS;

DIFFERENTIAL EQUATIONS; DYNAMICAL SYSTEMS; EIGENVALUES AND EIGENFUNCTIONS; LINEARIZATION; LYAPUNOV FUNCTIONS; LYAPUNOV METHODS; QUANTUM ELECTRONICS; QUANTUM OPTICS; STRUCTURAL MEMBERS; SWITCHING SYSTEMS; TRAJECTORIES;

ASYMPTOTIC ANALYSIS;

EID: 77957704892     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2010.2043292     Document Type: Article

References (27)

1. P. Vettori, "On the convergence of a feedback control strategy for multilevel quantum systems," presented at the MTNS'02, 2002, paper no. 21350.
2. A. Ferrante, M. Pavon, and G. Raccanelli, "Driving the propagator of a spin system: A feedback approach," in Proc. 41st IEEE CDC'02, 2002, pp. 46-50.
3. S. Grivopoulos and B. Bamieh, "Lyapunov-based control of quantum systems," in Proc. 42nd IEEE CDC'03, 2003, pp. 434-438.
4. M. Mirrahimi and P. Rouchon, "Trajectory generation for quantum systems based on Lyapunov techniques," in Proc. NOLCOS'04, 2004, p. 291.
5. M. Mirrahimi and P. Rouchon, "Trajectory tracking for quantum systems: A Lyapunov approach," in Proc. MTNS'04, 2004, [CD ROM].
6. M. Mirrahimi and G. Turinici, "Lyapunov control of bilinear Schrodinger equations," Automatica, vol. 41, pp. 1987-1994, 2005.
7. K. Beauchard, J.-M. Coron, M. Mirrahimi, and P. Rouchon, "Implicit Lyapunov control of finite dimensional Schrodinger equations," Syst. Control Lett., vol. 56, pp. 388-395, 2007.
8. M. Mirrahimi and R. Van Handel, "Stabilizing feedback controls for quantum systems," SIAM J. Control Optim., vol. 46, pp. 445-467, 2007.
9. C. Altafini, "Feedback control of spin systems," Quantum Inform. Processing, vol. 6, pp. 9-36, 2007.
10. C. Altafini, "Feedback stabilization of quantum ensembles: A global convergence analysis on complex flag manifolds," IEEE Trans. Autom. Control, vol. 52, pp. 2019-2031, 2007.
11. D. D'Alessandro, Introduction to Quantum Control and Dynamics.: CRC Press, 2007.
12. V. Jurdjevic and J. P. Quinn, "Controllability and stability," J. Differential Equations, vol. 28, pp. 381-389, 1978.
13. H.-P. Breuer and F. Petrucione, Theory of Open Quantum Systems.: Oxford University Press, 2002.
14. R. Wu, H. Rabitz, and M. Hsieh, "Characterization of the critical sub-manifolds in the quantum ensemble control landscapes," J. Phys. A, vol. 41, p. 015006, 2008.
15. J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method With Applications. New York: Academic Press, 1961.
16. M. D. Girardeau et al., "Kinematical bounds on evolution and optimization of mixed quantum states," Phys. Rev. A, vol. 55, pp. R1565-R1568, 1998.
17. S. G. Schirmer and J. V. Leahy, "Limits of control for quantum systems: Kinematical bounds on the optimization of observables and the question of dynamical reachability," Phys. Rev. A, vol. 63, p. 025403, 2001.
18. S. Bohacek and E. A. Jonckheere, "Linear dynamically varying LQ control of nonlinear systems over compact sets," IEEE Trans. Autom. Control, vol. 46, no. 6, pp. 840-852, Jun. 2001.
19. N. Jacobson, Lie Algebras. New York: Dover, 1979.
20. F. Silva Leite and P. E. Crouch, "Controllability on classical Lie groups. Mathematics of Control," Signals Syst., vol. 1, pp. 31-42, 1988.
21. C. Altafini, "Controllability of quantum-mechanical systems by root space decomposition of su(n)," J. Math. Phys., vol. 43, pp. 2051-2062, 2002.
22. Y. Matsumoto, An Introduction to Morse Theory. Providence, RI: American Mathematical Society, 2002.
23. P. Glendinning, Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations. Cambridge, U.K.: Cambridge Univ. Press, 1994.
24. L. Perko, Differential Equations and Dynamical Systems. New York: Springer Verlag, 2000.
25. D. A. Harville, Matrix Algebra From a Statistician's Perspective. New York: Springer-Verlag, 1997.
26. J. Carr, Applications of Centre Manifold Theory. New York: Springer-Verlag, 1981.
27. U. Helmke and J.B. Moore, Optimization and Dynamical Systems. London, U.K.: Springer-Verlag, 1996.