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Constrained minimum-energy optimal control of the dissipative Bloch equations. (English) Zbl 1201.49021

Summary: We apply optimal control theory to design minimum-energy \(\pi /2\) and \(\pi \) pulses for the Bloch system in the presence of relaxation with constrained control amplitude. We consider a commonly encountered case in which the transverse relaxation rate is much larger than the longitudinal one so that the latter can be neglected. Using Pontryagin’s maximum principle, we derive optimal feedback laws which are characterized by the number of switches, depending on the control bound and the coordinates of the desired final state.

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
49M05 Numerical methods based on necessary conditions
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[1] Pontryagin, L. S.; Boltyanskii, V. G.; Gamkrelidze, R. V.; Mishchenko, E. F., The Mathematical Theory of Optimal Processes (1962), Interscience Publishers: Interscience Publishers New York · Zbl 0102.32001
[2] Khaneja, N.; Reiss, T.; Luy, B.; Glasser, S. J., Optimal control of spin dynamics in the presence of relaxation, J. Magn. Reson., 162, 311-319 (2003)
[3] Khaneja, N.; Luy, B.; Glaser, S. J., Boundary of quantum evolution under decoherence, Proc. Natl. Acad. Sci., 100, 13162-13166 (2003) · Zbl 1064.81018
[4] Khaneja, N.; Li, J.-S.; Kehlet, C.; Luy, B.; Glaser, S. J., Broadband relaxation-optimized polarization transfer in magnetic resonance, P. Natl. Acad. Sci. USA, 101, 14742-14747 (2004)
[5] Frueh, D. P.; Ito, T.; Li, J.-S.; Wagner, G.; Glaser, S. J.; Khaneja, N., Sensitivity enhancement in NMR of macromolecules by application of optimal control theory, J. Biomol. NMR, 32, 23-30 (2005)
[6] Stefanatos, D.; Khaneja, N.; Glaser, S. J., Optimal control of coupled spins in the presence of longitudinal and transverse relaxation, Phys. Rev. A, 69, 022319 (2004)
[7] Stefanatos, D.; Glaser, S. J.; Khaneja, N., Relaxation-optimized transfer of spin order in Ising spin chains, Phys. Rev. A, 72, 062320 (2005)
[8] Sugny, D.; Kontz, C.; Jauslin, H. R., Time-optimal control of a two-level dissipative quantum system, Phys. Rev. A, 76, 023419 (2007)
[9] Bonnard, B.; Sugny, D., Time-minimal control of dissipative two-level quantum systems: the integrable case, SIAM, J. Control Optim., 48, 3, 1289-1308 (2009) · Zbl 1288.81052
[10] Bonnard, B.; Chyba, M.; Sugny, D., Time-minimal control of dissipative two-level quantum systems: the generic case, IEEE Trans. Automat. Control., 54, 11, 2598-2610 (2009) · Zbl 1367.81072
[11] Wang, L. C.; Huang, X. L.; Yi, X. X., Effect of feedback on the control of a two-level dissipative quantum system, Phys. Rev. A, 78, 052112 (2008)
[12] Stefanatos, D., Optimal design of minimum-energy pulses for Bloch equations in the case of dominant transverse relaxation, Phys. Rev. A, 80, 045401 (2009)
[13] B. Bonnard, O. Cots, N. Shcherbakova, D. Sugny, The energy minimization problem for two-level dissipative quantum systems, preprint IMB 581 (2009), http://math.u-bourgogne.fr/IMB/; B. Bonnard, O. Cots, N. Shcherbakova, D. Sugny, The energy minimization problem for two-level dissipative quantum systems, preprint IMB 581 (2009), http://math.u-bourgogne.fr/IMB/ · Zbl 1309.81118
[14] Li, J.-S.; Ruths, J.; Stefanatos, D., A pseudospectral method for optimal control of open quantum systems, J. Chem. Phys., 131, 164110 (2009)
[15] D’Alessandro, D.; Dahleh, M., Optimal control of two-level quantum systems, IEEE Trans. Automat. Control., 46, 6, 866-876 (2001) · Zbl 0993.81070
[16] Boscain, U.; Mason, P., Time minimal trajectories for a spin-1/2 particle in a magnetic field, J. Math. Phys., 47, 062101 (2006) · Zbl 1112.81098
[17] Ernst, R. R.; Bodenhausen, G.; Wokaun, A., Principles of Nuclear Magnetic Resonance in One and Two Dimensions (1987), Clarendon Press: Clarendon Press Oxford
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