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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

Bernard Bonnard 1, , Monique Chyba 2, , Alain Jacquemard 1, and  John Marriott 2,

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.
Keywords: contrast imaging, Mayer problem, geometric optimal control, invariant theory..
Mathematics Subject Classification: 14L24, 49K15, 81Q9.
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 & 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], (1980).   Google Scholar

[2]

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

[3]

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

[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.  doi: 10.1137/110833427.  Google Scholar

[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., ().   Google Scholar

[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.  doi: 10.1109/TAC.2012.2195859.  Google Scholar

[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.  doi: 10.1515/form.1993.5.111.  Google Scholar

[8]

D. Cox, J. Little and D. O'Shea, "Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra,'', Third edition, (2007).  doi: 10.1007/978-0-387-35651-8.  Google Scholar

[9]

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

[10]

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

[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.  doi: 10.1016/S0747-7171(88)80040-3.  Google Scholar

[12]

P. Hartman, "Ordinary Differential Equations,'', Classics in Applied Mathematics, 38 (2002).  doi: 10.1137/1.9780898719222.  Google Scholar

[13]

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

[14]

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

[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.  doi: 10.1016/j.jmr.2004.11.004.  Google Scholar

[16]

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

[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).   Google Scholar

[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).  doi: 10.1038/srep00589.  Google Scholar

[19]

T. Levi-Civita, "The Absolute Differential Calculus. Calculus of Tensors,'', Reprint of the 1926 translation, (1926).   Google Scholar

[20]

M. H. Levitt, "Spin Dynamics: Basics of Nuclear Magnetic Resonance,'', Second edition, (2008).   Google Scholar

[21]

L. Markus, Quadratic differential equations and non-associative algebras,, in, (1960), 185.   Google Scholar

[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.  doi: 10.1016/0899-8248(89)90027-X.  Google Scholar

[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 (1994).  doi: 10.1007/978-3-642-57916-5.  Google Scholar

[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, (1962).   Google Scholar

[25]

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

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], (1980).   Google Scholar

[2]

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

[3]

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

[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.  doi: 10.1137/110833427.  Google Scholar

[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., ().   Google Scholar

[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.  doi: 10.1109/TAC.2012.2195859.  Google Scholar

[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.  doi: 10.1515/form.1993.5.111.  Google Scholar

[8]

D. Cox, J. Little and D. O'Shea, "Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra,'', Third edition, (2007).  doi: 10.1007/978-0-387-35651-8.  Google Scholar

[9]

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

[10]

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

[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.  doi: 10.1016/S0747-7171(88)80040-3.  Google Scholar

[12]

P. Hartman, "Ordinary Differential Equations,'', Classics in Applied Mathematics, 38 (2002).  doi: 10.1137/1.9780898719222.  Google Scholar

[13]

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

[14]

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

[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.  doi: 10.1016/j.jmr.2004.11.004.  Google Scholar

[16]

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

[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).   Google Scholar

[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).  doi: 10.1038/srep00589.  Google Scholar

[19]

T. Levi-Civita, "The Absolute Differential Calculus. Calculus of Tensors,'', Reprint of the 1926 translation, (1926).   Google Scholar

[20]

M. H. Levitt, "Spin Dynamics: Basics of Nuclear Magnetic Resonance,'', Second edition, (2008).   Google Scholar

[21]

L. Markus, Quadratic differential equations and non-associative algebras,, in, (1960), 185.   Google Scholar

[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.  doi: 10.1016/0899-8248(89)90027-X.  Google Scholar

[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 (1994).  doi: 10.1007/978-3-642-57916-5.  Google Scholar

[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, (1962).   Google Scholar

[25]

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

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