December  2013, 3(4): 375-396. doi: 10.3934/mcrf.2013.3.375

On the application of geometric optimal control theory to Nuclear Magnetic Resonance

1. 

Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 5209 CNRS-Université de Bourgogne, 9 Av. A. Savary, BP 47 870, F-21078 DIJON Cedex, France, France, France

2. 

Department of Chemistry, Technische Universität München, Lichtenbergstrasse 4, D-85747 Garching, Germany

Received  October 2012 Revised  April 2013 Published  September 2013

We present some applications of geometric optimal control theory to control problems in Nuclear Magnetic Resonance (NMR) and Magnetic Resonance Imaging (MRI). Using the Pontryagin Maximum Principle (PMP), the optimal trajectories are found as solutions of a pseudo-Hamiltonian system. This computation can be completed by second-order optimality conditions based on the concept of conjugate points. After a brief physical introduction to NMR, this approach is applied to analyze two relevant optimal control issues in NMR and MRI: the control of a spin 1/2 particle in presence of radiation damping effect and the maximization of the contrast in MRI. The theoretical analysis is completed by numerical computations. This work has been made possible by the central and essential role of B. Bonnard, who has been at the heart of this project since 2009.
Citation: Elie Assémat, Marc Lapert, Dominique Sugny, Steffen J. Glaser. On the application of geometric optimal control theory to Nuclear Magnetic Resonance. Mathematical Control and Related Fields, 2013, 3 (4) : 375-396. doi: 10.3934/mcrf.2013.3.375
References:
[1]

K. Beauchard, J.-M. Coron and P. Rouchon, Controllability issues for continuous-spectrum systems and ensemble controllability of Bloch equations, Comm. Math. Phys., 296 (2010), 525-557. doi: 10.1007/s00220-010-1008-9.

[2]

M. A. Bernstein, K. F. King and X. J. Zhou, Handbook of MRI pulse sequences, Medical Physics, 32 (2005), 1452. doi: 10.1118/1.1904597.

[3]

A. Bhattacharya, Chemistry: Breaking the billion-hertz barrier, Nature, 463 (2010), 605-606. doi: 10.1038/463605a.

[4]

N. Bloembergen and R. V. Pound, Radiation damping in magnetic resonance experiments, Phys. Rev., 95 (1954), 8-12. doi: 10.1103/PhysRev.95.8.

[5]

B. Bonnard, J.-B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control, ESAIM COCV, 13 (2007), 207-236. doi: 10.1051/cocv:2007012.

[6]

B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Conjugate and cut loci of a two-sphere of revolution with application to optimal control, Ann. Inst. H. Poincaré Anal. Non linéaire, 26 (2009), 1081-1098. doi: 10.1016/j.anihpc.2008.03.010.

[7]

B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory," Mathématiques & Applications (Berlin) [Mathematics & Applications], 40, Springer-Verlag, Berlin, 2003.

[8]

B. Bonnard, M. Chyba and J. Marriott, Singular trajectories and the contrast imaging problem in nuclear magnetic resonance, SIAM, J. Cont. Optim., 51 (2013), 1325-1349. doi: 10.1137/110833427.

[9]

B. Bonnard, M. Chyba and D. Sugny, Time-minimal control of dissipative two-level quantum systems: The generic case, IEEE Trans. Auto. Control, 54 (2009), 2598-2610. doi: 10.1109/TAC.2009.2031212.

[10]

B. Bonnard, M. Claeys, O. Cots and P. Martinon, Comparison of numerical methods in the contrast imaging problem in NMR, submitted to IEEE Trans. Auto. Control, (2013).

[11]

B. Bonnard, O. Cots, S. J. Glaser, M. Lapert, D. Sugny and Y. Zhang, Geometric optimal control of the contrast imaging problem in nuclear magnetic resonance, IEEE Transactions on Automatic and Control, 57 (2012), 1957-1969. doi: 10.1109/TAC.2012.2195859.

[12]

B. Bonnard, O. Cots, N. Shcherbakova and D. Sugny, The energy minimization problem for two-level dissipative quantum systems, J. Math. Phys., 51 (2010), 092705, 44 pp. doi: 10.1063/1.3479390.

[13]

B. Bonnard and D. Sugny, Time minimal control of dissipative two-level quantum systems: The integrable case, SIAM, J. Cont. Optim., 48 (2009), 1289-1308. doi: 10.1137/080717043.

[14]

U. Boscain and P. Mason, Time minimal trajectories for a spin 1/2 particle in a magnetic field, J. Math. Phys., 47 (2006), 062101, 29 pp. doi: 10.1063/1.2203236.

[15]

D. O. Brunner, N. De Zanche, J. Fröhlich, J. Paska and K. P. Pruessmann, Travelling-wave nuclear magnetic resonance, Nature, 457 (2009), 994-998. doi: 10.1038/nature07752.

[16]

G. M. Bydder, J. V. Hajnal and I. R. Young, MRI: Use of the inversion recovery pulse sequence, Clinical Radiology, 53 (1998), 159-176. doi: 10.1016/S0009-9260(98)80096-2.

[17]

G. M. Bydder and I. R. Yound, MR imaging: Clinical use of the inversion recovery sequence, Journal of Computed Assisted Tomography, 9 (1985). doi: 10.1097/00004728-198507010-00002.

[18]

S. Conolly, D. Nishimura and A. Macovski, Optimal control solutions to the magnetic resonance selective excitation problem, IEEE Trans. Med. Imaging, 5 (1986), 106-115. doi: 10.1109/TMI.1986.4307754.

[19]

O. Cots, "Contrôle Optimal Géométrique: Méthodes Homotopiques et Applications," Ph.D thesis, Univ. Bourgogne, 2012.

[20]

D. D'Alessandro and M. Dahleh, Optimal control of two-level quantum systems, IEEE Trans. Auto. Control, 46 (2001), 866-876. doi: 10.1109/9.928587.

[21]

R. R. Ernst, G. Bodenhausen and A. Wokaun, "Principles of Nuclear Magnetic Resonance in One and Two Dimensions," International Series of Monographs on Chemistry, Oxford University Press, Oxford, 1990. doi: 10.1063/1.2811094.

[22]

L. Frydman and D. Blazina, Ultrafast two-dimensional nuclear magnetic resonance spectroscopy of hyperpolarized solutions, Nature Physics, 3 (2007), 415-419. doi: 10.1038/nphys597.

[23]

N. I. Gershenzon, K. Kobzar, B. Luy, S. J. Glaser and T. E. Skinner, Optimal control design of excitation pulses that accomodate relaxation, J. Magn. Reson., 188 (2007), 330-336. doi: 10.1016/j.jmr.2007.08.007.

[24]

V. Jurdjevic, "Geometric Control Theory," Cambridge Studies in Advanced Mathematics, 52, Cambridge University Press, Cambridge, 1997.

[25]

N. Khaneja, R. Brockett and S. J. Glaser, Time optimal control in spin systems, Phys. Rev. A, 63 (2001), 032308, 13 pp. doi: 10.1103/PhysRevA.63.032308.

[26]

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 (3), 65 (2002), 032301, 11 pp. doi: 10.1103/PhysRevA.65.032301.

[27]

N. Khaneja, F. Kramer and S. J. Glaser, Optimal experiments for maximizing coherence transfer between coupled spins, J. Magn. Reson., 173 (2005), 116-124. doi: 10.1016/j.jmr.2004.11.023.

[28]

N. Khaneja, B. Luy and S. J. Glaser, Boundary of quantum evolution under decoherence, Proc. Natl. Acad. Sci. USA, 100 (2003), 13162-13166. doi: 10.1073/pnas.2134111100.

[29]

N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen and S. J. Glaser, Optimal control of coupled spin dynamics: Design of NMR pulse sequences by gradient ascent algorithms, J. Magn. Reson., 172 (2005), 296-305. doi: 10.1016/j.jmr.2004.11.004.

[30]

N. Khaneja, T.Reiss, B. Luy and S. J. Glaser, Optimal control of spin dynamics in the presence of relaxation, J. Magn. Reson., 162 (2003), 311-319. doi: 10.1016/S1090-7807(03)00003-X.

[31]

K. Kobzar, B. Luy, N. Khaneja and S. J. Glaser, Pattern pulses: Design of arbitrary excitation profiles as a function of pulse amplitude and offset, J. Magn. Reson., 173 (2005), 229-235. doi: 10.1016/j.jmr.2004.12.005.

[32]

K. Kobzar, T. E. Skinner, N. Khaneja, S. J. Glaser and B. Luy, Exploring the limits of broadband excitation and inversion pulses, J. Magn. Reson., 170 (2004), 236-243. doi: 10.1016/j.jmr.2004.06.017.

[33]

K. Kobzar, T. E. Skinner, N. Khaneja, S. J. Glaser and B. Luy, Exploring the limits of broadband excitation and inversion: II. Rf-power optimized pulses, J. Magn. Reson., 194 (2008), 58-66. doi: 10.1016/j.jmr.2008.05.023.

[34]

M. Lapert, Y. Zhang, M. Braun, S. J. Glaser and D. Sugny, Singular extremals for the time-optimal control of dissipative spin 1/2 particles, Phys. Rev. Lett., 104 (2010), 083001, 4 pp. doi: 10.1103/PhysRevLett.104.083001.

[35]

M. Lapert, Y. Zhang, M. Janich, S. J. Glaser and D. Sugny, Exploring the physical limits of contrast in magnetic resonance imaging, Sci. Rep., 2 (2012), Art. Num. 589. doi: 10.1038/srep00589.

[36]

M. H. Levitt, "Spin Dynamics: Basics of Nuclear Magnetic Resonance," John Wiley & sons, New York-London-Sydney, 2008. doi: 10.1118/1.3273534.

[37]

J.-S. Li and N. Khaneja, Ensemble controllability of the Bloch equations, in "2006 45th IEEE Conference on Decision and Control," (2006), 2483-2487.

[38]

J. Mao, T. H. Mareci, K. N. Scott and E. R. Andrew, Selective inversion radiofrequency pulses by optimal control, J. Magn. Reson., 70 (1986), 310-318. doi: 10.1016/0022-2364(86)90016-8.

[39]

N. C. Nielsen, C. Kehlet, S. J. Glaser and N. Khaneja, "Optimal Control Methods in NMR Spectroscopy," Encyclopedia of Nuclear Magnetic Resonance, Wiley, 2010.

[40]

G. Pileio, M. Carravetta and M. H. Levitt, Extremely low-frequency spectroscopy in low-field nuclear magnetic resonance, Phys. Rev. Lett., 103 (2009), 083002, 4 pp. doi: 10.1103/PhysRevLett.103.083002.

[41]

L. Pontryagin, et al., "Mathematical Theory of Optimal Processes," Mir, Moscou, 1974.

[42]

T. O. Reiss, N. Khaneja and S. J. Glaser, Broadband geodesic pulses for three spin systems: Time-optimal realization of effective trilinear coupling terms and indirect SWAP gates, J. Magn. Reson., 165 (2003), 95-101. doi: 10.1016/S1090-7807(03)00245-3.

[43]

S. Rice and M. Zhao, "Optimal Control of Molecular Dynamics," Wiley, New York, 2000.

[44]

D. Rosenfeld and Y. Zun, Design of adiabatic selective pulses using optimal control theory, Magn. Reson. Med., 36 (1996), 401-409. doi: 10.1002/mrm.1910360311.

[45]

J. N. Rydberg, S. J. Riederer, C. H. Rydberg and C. R. Jack, Contrast optimization of fluid-attenuated inversion recovery (flair) imaging, Magnetic Resonance in Medicine, 34 (1995), 868-877. doi: 10.1002/mrm.1910340612.

[46]

T. E. Skinner, T. O. Reiss, B. Luy, N. Khaneja and S. J. Glaser, Application of optimal control theory to the design of broadband excitation pulses for high resolution NMR, J. Magn. Reson., 163 (2003), 8-15. doi: 10.1016/S1090-7807(03)00153-8.

[47]

D. Stefanatos, N. Khaneja and S. J. Glaser, Optimal control of coupled spins in presence of longitudinal and transverse relaxation, Phys. Rev. A, 69 (2004), 022319.

[48]

D. Stefanatos, N. Khaneja and S. J. Glaser, Relaxation optimized transfer of spin order in ising chains, Phys. Rev. A, 72 (2005), 062320. doi: 10.1103/PhysRevA.72.062320.

[49]

D. J. Tannor, "Introduction to Quantum Mechanics: A Time-Dependent Perspective," University Science Books, Sausalito, California, 2007.

[50]

Z. Tosner, T. Vosegaard, C. Kehlet, N. Khaneja, S. J. Glaser and N. C. Nielsen, Optimal control in NMR spectroscopy: Numerical implementation in SIMPSON, J. Magn. Reson., 197 (2009), 120-134.

[51]

L. M. K. Vandersypen and I. L. Chuang, NMR techniques for quantum control and computation, Rev. Mod. Phys., 76 (2004), 1037-1069.

[52]

M. S. Vinding, I. I. Maximov, Z. Tosner and N. C. Nielsen, Fast numerical design of spatil-selective rf pulses in MRI using Krotov and quasi-Newton based optimal control methods, J. Chem. Phys., 137 (2012), 054203. doi: 10.1063/1.4739755.

[53]

W. S. Warren, S. L. Hammes and J. L. Bates, Dynamics of radiation damping in nuclear magnetic resonance, J. Chem. Phys., 91 (1989), 5895. doi: 10.1063/1.457458.

[54]

Y. Zhang, M. Lapert, M. Braun, D. Sugny and S. J. Glaser, Time-optimal control of spin 1/2 particles in presence of relaxation and radiation damping effects, J. Chem. Phys., 134 (2011), 054103.

show all references

References:
[1]

K. Beauchard, J.-M. Coron and P. Rouchon, Controllability issues for continuous-spectrum systems and ensemble controllability of Bloch equations, Comm. Math. Phys., 296 (2010), 525-557. doi: 10.1007/s00220-010-1008-9.

[2]

M. A. Bernstein, K. F. King and X. J. Zhou, Handbook of MRI pulse sequences, Medical Physics, 32 (2005), 1452. doi: 10.1118/1.1904597.

[3]

A. Bhattacharya, Chemistry: Breaking the billion-hertz barrier, Nature, 463 (2010), 605-606. doi: 10.1038/463605a.

[4]

N. Bloembergen and R. V. Pound, Radiation damping in magnetic resonance experiments, Phys. Rev., 95 (1954), 8-12. doi: 10.1103/PhysRev.95.8.

[5]

B. Bonnard, J.-B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control, ESAIM COCV, 13 (2007), 207-236. doi: 10.1051/cocv:2007012.

[6]

B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Conjugate and cut loci of a two-sphere of revolution with application to optimal control, Ann. Inst. H. Poincaré Anal. Non linéaire, 26 (2009), 1081-1098. doi: 10.1016/j.anihpc.2008.03.010.

[7]

B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory," Mathématiques & Applications (Berlin) [Mathematics & Applications], 40, Springer-Verlag, Berlin, 2003.

[8]

B. Bonnard, M. Chyba and J. Marriott, Singular trajectories and the contrast imaging problem in nuclear magnetic resonance, SIAM, J. Cont. Optim., 51 (2013), 1325-1349. doi: 10.1137/110833427.

[9]

B. Bonnard, M. Chyba and D. Sugny, Time-minimal control of dissipative two-level quantum systems: The generic case, IEEE Trans. Auto. Control, 54 (2009), 2598-2610. doi: 10.1109/TAC.2009.2031212.

[10]

B. Bonnard, M. Claeys, O. Cots and P. Martinon, Comparison of numerical methods in the contrast imaging problem in NMR, submitted to IEEE Trans. Auto. Control, (2013).

[11]

B. Bonnard, O. Cots, S. J. Glaser, M. Lapert, D. Sugny and Y. Zhang, Geometric optimal control of the contrast imaging problem in nuclear magnetic resonance, IEEE Transactions on Automatic and Control, 57 (2012), 1957-1969. doi: 10.1109/TAC.2012.2195859.

[12]

B. Bonnard, O. Cots, N. Shcherbakova and D. Sugny, The energy minimization problem for two-level dissipative quantum systems, J. Math. Phys., 51 (2010), 092705, 44 pp. doi: 10.1063/1.3479390.

[13]

B. Bonnard and D. Sugny, Time minimal control of dissipative two-level quantum systems: The integrable case, SIAM, J. Cont. Optim., 48 (2009), 1289-1308. doi: 10.1137/080717043.

[14]

U. Boscain and P. Mason, Time minimal trajectories for a spin 1/2 particle in a magnetic field, J. Math. Phys., 47 (2006), 062101, 29 pp. doi: 10.1063/1.2203236.

[15]

D. O. Brunner, N. De Zanche, J. Fröhlich, J. Paska and K. P. Pruessmann, Travelling-wave nuclear magnetic resonance, Nature, 457 (2009), 994-998. doi: 10.1038/nature07752.

[16]

G. M. Bydder, J. V. Hajnal and I. R. Young, MRI: Use of the inversion recovery pulse sequence, Clinical Radiology, 53 (1998), 159-176. doi: 10.1016/S0009-9260(98)80096-2.

[17]

G. M. Bydder and I. R. Yound, MR imaging: Clinical use of the inversion recovery sequence, Journal of Computed Assisted Tomography, 9 (1985). doi: 10.1097/00004728-198507010-00002.

[18]

S. Conolly, D. Nishimura and A. Macovski, Optimal control solutions to the magnetic resonance selective excitation problem, IEEE Trans. Med. Imaging, 5 (1986), 106-115. doi: 10.1109/TMI.1986.4307754.

[19]

O. Cots, "Contrôle Optimal Géométrique: Méthodes Homotopiques et Applications," Ph.D thesis, Univ. Bourgogne, 2012.

[20]

D. D'Alessandro and M. Dahleh, Optimal control of two-level quantum systems, IEEE Trans. Auto. Control, 46 (2001), 866-876. doi: 10.1109/9.928587.

[21]

R. R. Ernst, G. Bodenhausen and A. Wokaun, "Principles of Nuclear Magnetic Resonance in One and Two Dimensions," International Series of Monographs on Chemistry, Oxford University Press, Oxford, 1990. doi: 10.1063/1.2811094.

[22]

L. Frydman and D. Blazina, Ultrafast two-dimensional nuclear magnetic resonance spectroscopy of hyperpolarized solutions, Nature Physics, 3 (2007), 415-419. doi: 10.1038/nphys597.

[23]

N. I. Gershenzon, K. Kobzar, B. Luy, S. J. Glaser and T. E. Skinner, Optimal control design of excitation pulses that accomodate relaxation, J. Magn. Reson., 188 (2007), 330-336. doi: 10.1016/j.jmr.2007.08.007.

[24]

V. Jurdjevic, "Geometric Control Theory," Cambridge Studies in Advanced Mathematics, 52, Cambridge University Press, Cambridge, 1997.

[25]

N. Khaneja, R. Brockett and S. J. Glaser, Time optimal control in spin systems, Phys. Rev. A, 63 (2001), 032308, 13 pp. doi: 10.1103/PhysRevA.63.032308.

[26]

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 (3), 65 (2002), 032301, 11 pp. doi: 10.1103/PhysRevA.65.032301.

[27]

N. Khaneja, F. Kramer and S. J. Glaser, Optimal experiments for maximizing coherence transfer between coupled spins, J. Magn. Reson., 173 (2005), 116-124. doi: 10.1016/j.jmr.2004.11.023.

[28]

N. Khaneja, B. Luy and S. J. Glaser, Boundary of quantum evolution under decoherence, Proc. Natl. Acad. Sci. USA, 100 (2003), 13162-13166. doi: 10.1073/pnas.2134111100.

[29]

N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen and S. J. Glaser, Optimal control of coupled spin dynamics: Design of NMR pulse sequences by gradient ascent algorithms, J. Magn. Reson., 172 (2005), 296-305. doi: 10.1016/j.jmr.2004.11.004.

[30]

N. Khaneja, T.Reiss, B. Luy and S. J. Glaser, Optimal control of spin dynamics in the presence of relaxation, J. Magn. Reson., 162 (2003), 311-319. doi: 10.1016/S1090-7807(03)00003-X.

[31]

K. Kobzar, B. Luy, N. Khaneja and S. J. Glaser, Pattern pulses: Design of arbitrary excitation profiles as a function of pulse amplitude and offset, J. Magn. Reson., 173 (2005), 229-235. doi: 10.1016/j.jmr.2004.12.005.

[32]

K. Kobzar, T. E. Skinner, N. Khaneja, S. J. Glaser and B. Luy, Exploring the limits of broadband excitation and inversion pulses, J. Magn. Reson., 170 (2004), 236-243. doi: 10.1016/j.jmr.2004.06.017.

[33]

K. Kobzar, T. E. Skinner, N. Khaneja, S. J. Glaser and B. Luy, Exploring the limits of broadband excitation and inversion: II. Rf-power optimized pulses, J. Magn. Reson., 194 (2008), 58-66. doi: 10.1016/j.jmr.2008.05.023.

[34]

M. Lapert, Y. Zhang, M. Braun, S. J. Glaser and D. Sugny, Singular extremals for the time-optimal control of dissipative spin 1/2 particles, Phys. Rev. Lett., 104 (2010), 083001, 4 pp. doi: 10.1103/PhysRevLett.104.083001.

[35]

M. Lapert, Y. Zhang, M. Janich, S. J. Glaser and D. Sugny, Exploring the physical limits of contrast in magnetic resonance imaging, Sci. Rep., 2 (2012), Art. Num. 589. doi: 10.1038/srep00589.

[36]

M. H. Levitt, "Spin Dynamics: Basics of Nuclear Magnetic Resonance," John Wiley & sons, New York-London-Sydney, 2008. doi: 10.1118/1.3273534.

[37]

J.-S. Li and N. Khaneja, Ensemble controllability of the Bloch equations, in "2006 45th IEEE Conference on Decision and Control," (2006), 2483-2487.

[38]

J. Mao, T. H. Mareci, K. N. Scott and E. R. Andrew, Selective inversion radiofrequency pulses by optimal control, J. Magn. Reson., 70 (1986), 310-318. doi: 10.1016/0022-2364(86)90016-8.

[39]

N. C. Nielsen, C. Kehlet, S. J. Glaser and N. Khaneja, "Optimal Control Methods in NMR Spectroscopy," Encyclopedia of Nuclear Magnetic Resonance, Wiley, 2010.

[40]

G. Pileio, M. Carravetta and M. H. Levitt, Extremely low-frequency spectroscopy in low-field nuclear magnetic resonance, Phys. Rev. Lett., 103 (2009), 083002, 4 pp. doi: 10.1103/PhysRevLett.103.083002.

[41]

L. Pontryagin, et al., "Mathematical Theory of Optimal Processes," Mir, Moscou, 1974.

[42]

T. O. Reiss, N. Khaneja and S. J. Glaser, Broadband geodesic pulses for three spin systems: Time-optimal realization of effective trilinear coupling terms and indirect SWAP gates, J. Magn. Reson., 165 (2003), 95-101. doi: 10.1016/S1090-7807(03)00245-3.

[43]

S. Rice and M. Zhao, "Optimal Control of Molecular Dynamics," Wiley, New York, 2000.

[44]

D. Rosenfeld and Y. Zun, Design of adiabatic selective pulses using optimal control theory, Magn. Reson. Med., 36 (1996), 401-409. doi: 10.1002/mrm.1910360311.

[45]

J. N. Rydberg, S. J. Riederer, C. H. Rydberg and C. R. Jack, Contrast optimization of fluid-attenuated inversion recovery (flair) imaging, Magnetic Resonance in Medicine, 34 (1995), 868-877. doi: 10.1002/mrm.1910340612.

[46]

T. E. Skinner, T. O. Reiss, B. Luy, N. Khaneja and S. J. Glaser, Application of optimal control theory to the design of broadband excitation pulses for high resolution NMR, J. Magn. Reson., 163 (2003), 8-15. doi: 10.1016/S1090-7807(03)00153-8.

[47]

D. Stefanatos, N. Khaneja and S. J. Glaser, Optimal control of coupled spins in presence of longitudinal and transverse relaxation, Phys. Rev. A, 69 (2004), 022319.

[48]

D. Stefanatos, N. Khaneja and S. J. Glaser, Relaxation optimized transfer of spin order in ising chains, Phys. Rev. A, 72 (2005), 062320. doi: 10.1103/PhysRevA.72.062320.

[49]

D. J. Tannor, "Introduction to Quantum Mechanics: A Time-Dependent Perspective," University Science Books, Sausalito, California, 2007.

[50]

Z. Tosner, T. Vosegaard, C. Kehlet, N. Khaneja, S. J. Glaser and N. C. Nielsen, Optimal control in NMR spectroscopy: Numerical implementation in SIMPSON, J. Magn. Reson., 197 (2009), 120-134.

[51]

L. M. K. Vandersypen and I. L. Chuang, NMR techniques for quantum control and computation, Rev. Mod. Phys., 76 (2004), 1037-1069.

[52]

M. S. Vinding, I. I. Maximov, Z. Tosner and N. C. Nielsen, Fast numerical design of spatil-selective rf pulses in MRI using Krotov and quasi-Newton based optimal control methods, J. Chem. Phys., 137 (2012), 054203. doi: 10.1063/1.4739755.

[53]

W. S. Warren, S. L. Hammes and J. L. Bates, Dynamics of radiation damping in nuclear magnetic resonance, J. Chem. Phys., 91 (1989), 5895. doi: 10.1063/1.457458.

[54]

Y. Zhang, M. Lapert, M. Braun, D. Sugny and S. J. Glaser, Time-optimal control of spin 1/2 particles in presence of relaxation and radiation damping effects, J. Chem. Phys., 134 (2011), 054103.

[1]

Bernard Bonnard, Monique Chyba, Alain Jacquemard, John Marriott. Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance. Mathematical Control and Related Fields, 2013, 3 (4) : 397-432. doi: 10.3934/mcrf.2013.3.397

[2]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations and Control Theory, 2022, 11 (2) : 347-371. doi: 10.3934/eect.2020110

[3]

Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control and Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021

[4]

Zhen Wu, Feng Zhang. Maximum principle for discrete-time stochastic optimal control problem and stochastic game. Mathematical Control and Related Fields, 2022, 12 (2) : 475-493. doi: 10.3934/mcrf.2021031

[5]

Xiao-Li Ding, Iván Area, Juan J. Nieto. Controlled singular evolution equations and Pontryagin type maximum principle with applications. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021059

[6]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control and Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195

[7]

Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control and Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61

[8]

Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control and Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022

[9]

Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial and Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067

[10]

Simon Hubmer, Andreas Neubauer, Ronny Ramlau, Henning U. Voss. On the parameter estimation problem of magnetic resonance advection imaging. Inverse Problems and Imaging, 2018, 12 (1) : 175-204. doi: 10.3934/ipi.2018007

[11]

Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174

[12]

Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161

[13]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control and Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018

[14]

Miaomiao Chen, Rong Yuan. Maximum principle for the optimal harvesting problem of a size-stage-structured population model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4619-4648. doi: 10.3934/dcdsb.2021245

[15]

Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004

[16]

H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77

[17]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581

[18]

Serge Nicaise, Simon Stingelin, Fredi Tröltzsch. Optimal control of magnetic fields in flow measurement. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 579-605. doi: 10.3934/dcdss.2015.8.579

[19]

H. O. Fattorini. The maximum principle in infinite dimension. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557

[20]

Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial and Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967

2021 Impact Factor: 1.141

Metrics

  • PDF downloads (129)
  • HTML views (0)
  • Cited by (6)

[Back to Top]