Switching control of closed quantum systems via the Lyapunov method. (English) Zbl 1269.93099
Summary: In this paper, we consider the state transfer problem for closed quantum systems under a degenerate case, where the linearized system around the target state is not controllable. It is known that the traditional Lyapunov control methods may fail to guarantee the convergence to the target state under the degenerate case. Hence, we propose to use multiple Lyapunov functions and design a switching control strategy to achieve more accurate state transfer. It is shown that the system can converge to the intersection of invariant sets including the target state. The explicit analysis of the convergence is provided to design the switching law. Moreover, the effectiveness of open-loop Lyapunov control is discussed. Simulation studies are presented to show the improved control performance.
MSC:
| 93D30 | Lyapunov and storage functions |
| 93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |
Keywords:
switching control; quantum system; Lyapunov method; LaSalle invariance principle; effectiveness of open-loop Lyapunov control; state transfer problem; linearized systemReferences:
| [1] | Altafini, C., Feedback stabilization of isospectral control systems on complex flag manifolds: application to quantum ensembles, IEEE Transactions on Automatic Control, 52, 2019-2028 (2007) · Zbl 1366.93562 |
| [2] | Baldi, S.; Battistelli, G.; Mosca, E.; Tesi, P., Multi-model unfalsified adaptive switching supervisory control, Automatica, 46, 249-259 (2010) · Zbl 1205.93005 |
| [3] | Beauchard, K.; Coron, J.; Mirrahimi, M.; Rouchon, P., Implicit Lyapunov control of finite dimensional Schrödinger equations, Systems & Control Letters, 56, 388-395 (2007) · Zbl 1110.81063 |
| [4] | Coron, J.; Grigoriu, A.; Lefter, C.; Turinici, G., Quantum control design by Lyapunov trajectory tracking for dipole and polarizability coupling, New Journal of Physics, 11, 105034 (2009) |
| [5] | D’Alessandro, D., Introduction to quantum control and dynamics (2007), Chapman & Hall, CRC |
| [6] | D’Alessandro, D., Constructive decomposition of the controllability Lie algebra for quantum systems, IEEE Transactions on Automatic Control, 55, 1416-1421 (2010) · Zbl 1368.81088 |
| [7] | Dong, D.; Chen, C.; Tarn, T.; Pechen, A.; Rabitz, H., Incoherent control of quantum systems with wavefunction-controllable subspaces via quantum reinforcement learning, IEEE Transactions on Systems, Man and Cybernetics, Part B (Cybernetics), 38, 957-962 (2008) |
| [8] | Dong, D.; Petersen, I., Controllability of quantum systems with switching control, International Journal of Control, 84, 37-46 (2010) · Zbl 1222.93111 |
| [9] | Dong, Y.; Sun, J., On hybrid control of a class of stochastic non-linear Markovian switching systems, Automatica, 44, 990-995 (2008) · Zbl 1283.93251 |
| [10] | Grivopoulos, S., & Bamieh, B. (2003). Lyapunov-based control of quantum systems. In Proceedings of 42nd IEEE conference on decision and controlVol. 1; Grivopoulos, S., & Bamieh, B. (2003). Lyapunov-based control of quantum systems. In Proceedings of 42nd IEEE conference on decision and controlVol. 1 |
| [11] | Gross, P.; Neuhauser, D.; Rabitz, H., Optimal control of unimolecular reactions in the collisional regime, Journal of Chemical Physics, 94, 1158 (1991) |
| [12] | Ho, T.; Rabitz, H., Accelerated monotonic convergence of optimal control over quantum dynamics, Physical Review E, 82, 26703 (2010) |
| [13] | Khaneja, N., Switched control of electron nuclear spin systems, Physical Review A, 76, 32326 (2007) |
| [14] | Kuang, S.; Cong, S., Lyapunov control methods of closed quantum, Automatica, 44, 98-108 (2008) · Zbl 1138.93040 |
| [15] | Lin, H.; Antsaklis, P., Switching stabilizability for continuous-time uncertain switched linear systems, IEEE Transactions on Automatic Control, 52, 633-646 (2007) · Zbl 1366.93580 |
| [16] | Lou, Y.; Cong, S.; Yang, J.; Kuang, S., Path programming control strategy of quantum state transfer (regular papers), IET Control Theory & Applications, 5, 291-298 (2011) |
| [17] | Mirrahimi, M.; Handel, R. V., Stabilizing feedback controls for quantum systems, SIAM Journal on Control and Optimization, 46, 445-467 (2007) · Zbl 1136.81342 |
| [18] | Mirrahimi, M.; Rouchon, P.; Turinici, G., Lyapunov control of bilinear Schrödinger equations, Automatica, 41, 1987-1994 (2005) · Zbl 1125.93466 |
| [19] | Mitra, A.; Rabitz, H., Identifying mechanisms in the control of quantum dynamics through Hamiltonian encoding, Physical Review A, 67, 033407 (2003) |
| [20] | Nielsen, M. A.; Chuang, I. L., Quantum computation and quantum information (2000), Cambridge University Press: Cambridge University Press Cambridge, Massachusetts · Zbl 1049.81015 |
| [21] | Nurdin, H. I.; James, M. R.; Petersen, I., Coherent quantum LQG control, Automatica, 45, 1837-1846 (2009) · Zbl 1185.49037 |
| [22] | Pechen, A.; Il’in, N.; Shuang, F.; Rabitz, H., Quantum control by von Neumann measurements, Physical Review A, 74, 052102 (2006) |
| [23] | Phan, M.; Rabitz, H., A self-guided algorithm for learning control of quantum-mechanical systems, Journal of Chemical Physics, 110, 34 (1999) |
| [24] | Ramakrishna, V.; Salapaka, M.; Dahleh, M.; Rabitz, H.; Peirce, A., Controllability of molecular systems, Physical Review A, 51, 960 (1995) |
| [25] | Santarelli, K. R., A switched state feedback law for the stabilization of LTI systems, IEEE Transactions on Automatic Control, 56, 998-1013 (2011) · Zbl 1368.93564 |
| [26] | Tersigni, S. H.; Gaspard, P.; Rice, S. A., On using shaped light pulses to control the selectivity of product formation in a chemical reaction: an application to a multiple level system, Journal of Chemical Physics, 93, 1670-1680 (1990) |
| [27] | Van Handel, R.; Stockton, J. K.; Mabuchi, H., Feedback control of quantum state reduction, IEEE Transactions on Automatic Control, 50, 768-780 (2005) · Zbl 1365.81068 |
| [28] | Wang, X. T.; Schirmer, S. G., Analysis of effectiveness of Lyapunov control for nongeneric quantum states, IEEE Transactions on Automatic Control, 55, 1406-1411 (2010) · Zbl 1368.93612 |
| [29] | Wang, X. T.; Schirmer, S. G., Analysis of Lyapunov method for control of quantum states, IEEE Transactions on Automatic Control, 55, 2259-2270 (2010) · Zbl 1368.81094 |
| [30] | Wang, J.; Wiseman, H., Feedback-stabilization of an arbitrary pure state of a two-level atom, Physical Review A, 64, 63810 (2001) |
| [31] | Yamamoto, N.; Tsumura, K.; Hara, S., Feedback control of quantum entanglement in a two-spin system, Automatica, 43, 981-992 (2007) · Zbl 1282.93130 |
| [32] | Ye, X., Switching adaptive output-feedback control of nonlinearly parametrized systems, Automatica, 41, 983-989 (2005) · Zbl 1091.93016 |
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.
