December  2013, 3(4): 397-432. doi: 10.3934/mcrf.2013.3.397

Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance

1. 

Institut de Mathématiques de Bourgogne, UMR CNRS 5584, Dijon F-21078, France, France

2. 

University of Hawai'i, 2565 McCarthy Mall, Honolulu, HI 96822, United States, United States

Received  December 2012 Revised  April 2013 Published  September 2013

The analysis of the contrast problem in NMR medical imaging is essentially reduced to the analysis of the so-called singular trajectories of the system modeling the problem: a coupling of two spin 1/2 control systems. They are solutions of a constraint Hamiltonian vector field and restricting the dynamics to the zero level set of the Hamiltonian they define a vector field on $B_1 \times B_2$, where $B_1$ and $B_2$ are the Bloch balls of the two spin particles. In this article we classify the behaviors of the solutions in relation with the relaxation parameters using the concept of feedback classification. The optimality status is analyzed using the feedback invariant concept of conjugate points.
Citation: 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
References:
[1]

V. Arnol'd, "Chapitres Supplémentaires de la Théorie des Équations Différentielles Ordinaires,'' (French) [Supplementary Chapters to the Theory of Ordinary Differential Equations], Translated from the Russian by Djilali Embarek, "Mir," Moscow, 1980.

[2]

B. Bonnard, Feedback equivalence for nonlinear systems and the time optimal control problem, SIAM J. Control Optim., 29 (1991), 1300-1321. doi: 10.1137/0329067.

[3]

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

[4]

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

[5]

B. Bonnard and O. Cots, Geometric numerical methods and results in the control imaging problem in nuclear magnetic resonance, to appear in Mathematical Models and Methods in Applied Sciences.

[6]

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 Trans. Automat. Control, 57 (2012), 1957-1969. doi: 10.1109/TAC.2012.2195859.

[7]

B. Bonnard and I. Kupka, Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal, Forum Math., 5 (1993), 111-159. doi: 10.1515/form.1993.5.111.

[8]

D. Cox, J. Little and D. O'Shea, "Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra,'' Third edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2007. doi: 10.1007/978-0-387-35651-8.

[9]

J. A. Dieudonné and J. B. Carrell, Invariant theory, old and new, Advances in Math., 4 (1970), 1-80.

[10]

D. Eisenbud, C. Huneke and W. Vasconcelos, Direct methods for primary decomposition, Invent. Math., 110 (1992), 207-235. doi: 10.1007/BF01231331.

[11]

P. Gianni, G. Trager and G. Zacharias, Gröbner bases and primary decompositions of polynomial ideals. Computational aspects of commutative algebra, J. Symbolic Comput., 6 (1988), 149-167. doi: 10.1016/S0747-7171(88)80040-3.

[12]

P. Hartman, "Ordinary Differential Equations,'' Classics in Applied Mathematics, 38, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898719222.

[13]

A. Jacquemard and M. A. Teixeira, Effective algebraic geometry and normal forms of reversible mappings, Revista Matemática Complutense, 15 (2002), 31-55.

[14]

B. Jakubczyk, Curvatures of single-input control systems, Control and Cybernetics, 38 (2009), 1375-1391.

[15]

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. of Magnetic Resonance, 172 (2005), 296-305. doi: 10.1016/j.jmr.2004.11.004.

[16]

A. J. Krener, The high order maximal principle and its application to singular extremals, SIAM J. Control Optimization, 15 (1977), 256-293. doi: 10.1137/0315019.

[17]

M. Lapert, Y. Zhang, M. Braun, S. J. Glaser and D. Sugny, Singular extremals for the time-optimal control of dissipative spin $\frac{1}{2}$ particles, Phys. Rev. Lett., 104 (2010), 083001.

[18]

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

[19]

T. Levi-Civita, "The Absolute Differential Calculus. Calculus of Tensors,'' Reprint of the 1926 translation, Dover Publications, Inc., Mineola, NY, 2005.

[20]

M. H. Levitt, "Spin Dynamics: Basics of Nuclear Magnetic Resonance,'' Second edition, John Wiley & Sons, 2008.

[21]

L. Markus, Quadratic differential equations and non-associative algebras, in "Contributions to the theory of nonlinear oscillations, Vol. V," Princeton Univ. Press, Princeton, N.J., (1960), 185-213.

[22]

H. Melenk, H. M. Möller and W. Neun, Symbolic solution of large stationary chemical kinetics problems, IMPACT of Computing in Science and Engineering, 1 (1989), 138-167. doi: 10.1016/0899-8248(89)90027-X.

[23]

D. Mumford, J. Fogarty and F. Kirwan, "Geometric Invariant Theory,'' Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-57916-5.

[24]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,'' Translated from the Russian by K. N. Trirogoff, edited by L. W. Neustadt, Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962.

[25]

A. O. Remizov, Implicit differential equations and vector fields with nonisolated singular points, Mat. Sb., 193 (2002), 105-124. doi: 10.1070/SM2002v193n11ABEH000693.

show all references

References:
[1]

V. Arnol'd, "Chapitres Supplémentaires de la Théorie des Équations Différentielles Ordinaires,'' (French) [Supplementary Chapters to the Theory of Ordinary Differential Equations], Translated from the Russian by Djilali Embarek, "Mir," Moscow, 1980.

[2]

B. Bonnard, Feedback equivalence for nonlinear systems and the time optimal control problem, SIAM J. Control Optim., 29 (1991), 1300-1321. doi: 10.1137/0329067.

[3]

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

[4]

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

[5]

B. Bonnard and O. Cots, Geometric numerical methods and results in the control imaging problem in nuclear magnetic resonance, to appear in Mathematical Models and Methods in Applied Sciences.

[6]

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 Trans. Automat. Control, 57 (2012), 1957-1969. doi: 10.1109/TAC.2012.2195859.

[7]

B. Bonnard and I. Kupka, Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal, Forum Math., 5 (1993), 111-159. doi: 10.1515/form.1993.5.111.

[8]

D. Cox, J. Little and D. O'Shea, "Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra,'' Third edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2007. doi: 10.1007/978-0-387-35651-8.

[9]

J. A. Dieudonné and J. B. Carrell, Invariant theory, old and new, Advances in Math., 4 (1970), 1-80.

[10]

D. Eisenbud, C. Huneke and W. Vasconcelos, Direct methods for primary decomposition, Invent. Math., 110 (1992), 207-235. doi: 10.1007/BF01231331.

[11]

P. Gianni, G. Trager and G. Zacharias, Gröbner bases and primary decompositions of polynomial ideals. Computational aspects of commutative algebra, J. Symbolic Comput., 6 (1988), 149-167. doi: 10.1016/S0747-7171(88)80040-3.

[12]

P. Hartman, "Ordinary Differential Equations,'' Classics in Applied Mathematics, 38, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898719222.

[13]

A. Jacquemard and M. A. Teixeira, Effective algebraic geometry and normal forms of reversible mappings, Revista Matemática Complutense, 15 (2002), 31-55.

[14]

B. Jakubczyk, Curvatures of single-input control systems, Control and Cybernetics, 38 (2009), 1375-1391.

[15]

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. of Magnetic Resonance, 172 (2005), 296-305. doi: 10.1016/j.jmr.2004.11.004.

[16]

A. J. Krener, The high order maximal principle and its application to singular extremals, SIAM J. Control Optimization, 15 (1977), 256-293. doi: 10.1137/0315019.

[17]

M. Lapert, Y. Zhang, M. Braun, S. J. Glaser and D. Sugny, Singular extremals for the time-optimal control of dissipative spin $\frac{1}{2}$ particles, Phys. Rev. Lett., 104 (2010), 083001.

[18]

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

[19]

T. Levi-Civita, "The Absolute Differential Calculus. Calculus of Tensors,'' Reprint of the 1926 translation, Dover Publications, Inc., Mineola, NY, 2005.

[20]

M. H. Levitt, "Spin Dynamics: Basics of Nuclear Magnetic Resonance,'' Second edition, John Wiley & Sons, 2008.

[21]

L. Markus, Quadratic differential equations and non-associative algebras, in "Contributions to the theory of nonlinear oscillations, Vol. V," Princeton Univ. Press, Princeton, N.J., (1960), 185-213.

[22]

H. Melenk, H. M. Möller and W. Neun, Symbolic solution of large stationary chemical kinetics problems, IMPACT of Computing in Science and Engineering, 1 (1989), 138-167. doi: 10.1016/0899-8248(89)90027-X.

[23]

D. Mumford, J. Fogarty and F. Kirwan, "Geometric Invariant Theory,'' Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-57916-5.

[24]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,'' Translated from the Russian by K. N. Trirogoff, edited by L. W. Neustadt, Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962.

[25]

A. O. Remizov, Implicit differential equations and vector fields with nonisolated singular points, Mat. Sb., 193 (2002), 105-124. doi: 10.1070/SM2002v193n11ABEH000693.

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