ESAIM: Control, Optimisation and Calculus of Variations

The aim of this article is to classify the singular trajectories associated with the optimal control problems of a pair of controlled Bloch equations. The motivation is to analyze the robustness of the optimal solutions to the contrast and the time-minimal saturation problem, in magnetic resonance imaging, with respect to the parameters and B 1 -inhomogeneity. For this purpose, we use various computer algebra algorithms and methods to study solutions of polynomial systems of equations and inequalities which are used for classification issues: Gröbner basis, cylindrical algebraic decomposition of semi-algebraic sets, Thom’s isotopy lemma. Résumé. L’objectif de cet article est de classifier les trajectoires singulières associées aux problèmes de contrôle optimaux d’une paire d’équations de Bloch contrôlées. La motivation est d’analyser la robustesse de la solution optimale du problème de contraste et de multisaturation en temps minimal, en imagerie par résonance magnétique nucléaire, par rapport aux paramètres et les inhomogénéités B 1 . On utilise le calcul symbolique pour étudier les solutions de systèmes polynomiaux d’équations et d’inéquations dans les problèmes de classification : base de Gröbner, décomposition algébrique cylindrique des ensembles semi-algébriques et le lemme d’isotopie de Thom. 1991 Mathematics Subject Classification. 49K15,14Q20,81Q93.

[1]  Helene Ratiney,et al.  Optimal control theory for applications in Magnetic Resonance Imaging , 2017, 1709.02751.

[2]  Denis Grenier,et al.  Optimal control design of preparation pulses for contrast optimization in MRI. , 2017, Journal of magnetic resonance.

[3]  Jean-Charles Faugère,et al.  Determinantal Sets, Singularities and Application to Optimal Control in Medical Imagery , 2016, ISSAC.

[4]  Changbo Chen,et al.  Quantifier elimination by cylindrical algebraic decomposition based on regular chains , 2014, J. Symb. Comput..

[5]  Hoon Hong,et al.  On using Lazard's projection in CAD construction , 2016, J. Symb. Comput..

[6]  Marina Weber,et al.  Using Algebraic Geometry , 2016 .

[7]  Mathieu Claeys,et al.  Geometric and Numerical Methods in the Contrast Imaging Problem in Nuclear Magnetic Resonance , 2015 .

[8]  Mohab Safey El Din,et al.  Probabilistic Algorithm for Polynomial Optimization over a Real Algebraic Set , 2013, SIAM J. Optim..

[9]  Bernard Bonnard,et al.  GEOMETRIC NUMERICAL METHODS AND RESULTS IN THE CONTRAST IMAGING PROBLEM IN NUCLEAR MAGNETIC RESONANCE , 2014 .

[10]  John Marriott,et al.  Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance , 2013 .

[11]  John Marriott,et al.  Singular Trajectories and the Contrast Imaging Problem in Nuclear Magnetic Resonance , 2013, SIAM J. Control. Optim..

[12]  Marc Moreno Maza,et al.  On Solving Parametric Polynomial Systems , 2012, Mathematics in Computer Science.

[13]  Olivier Cots.  Contrôle optimal géométrique : méthodes homotopiques et applications , 2012 .

[14]  D. Sugny,et al.  Exploring the Physical Limits of Saturation Contrast in Magnetic Resonance Imaging , 2012, Scientific Reports.

[15]  Yun Zhang,et al.  Geometric Optimal Control of the Contrast Imaging Problem in Nuclear Magnetic Resonance , 2012, IEEE Transactions on Automatic Control.

[16]  Jean-Charles Faugère,et al.  Critical points and Gröbner bases: the unmixed case , 2012, ISSAC.

[17]  Kristian Ranestad.  Algebraic Degree in Semidefinite and Polynomial Optimization , 2012 .

[18]  M. Lapert,et al.  Développement de nouvelles techniques de contrôle optimal en dynamique quantique : de la Résonance Magnétique Nucléaire à la physique moléculaire , 2011 .

[19]  Changbo Chen,et al.  Algorithms for computing triangular decompositions of polynomial systems , 2011, ISSAC '11.

[20]  Jean-Charles Faugère,et al.  FGb: A Library for Computing Gröbner Bases , 2010, ICMS.

[21]  Y Zhang,et al.  Singular extremals for the time-optimal control of dissipative spin 1/2 particles. , 2010, Physical review letters.

[22]  Mohab Safey El Din,et al.  Testing Sign Conditions on a Multivariate Polynomial and Applications , 2007, Math. Comput. Sci..

[23]  Bican Xia,et al.  A complete algorithm for automated discovering of a class of inequality-type theorems , 2001, Science in China Series F Information Sciences.

[24]  Michael E. Taylor,et al.  Differential Geometry I , 1994 .

[25]  Timo O. Reiss,et al.  Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms. , 2005, Journal of magnetic resonance.

[26]  M. Chyba,et al.  Singular Trajectories and Their Role in Control Theory , 2003, IEEE Transactions on Automatic Control.

[27]  Alicia Dickenstein,et al.  Introduction to residues and resultants , 2005 .

[28]  Teresa Krick.  Solving Polynomial Equations , 2005 .

[29]  Christopher W. Brown.  QEPCAD B: a program for computing with semi-algebraic sets using CADs , 2003, SIGS.

[30]  Éric Schost,et al.  Polar varieties and computation of one point in each connected component of a smooth real algebraic set , 2003, ISSAC '03.

[31]  Timo O. Reiss,et al.  Application of optimal control theory to the design of broadband excitation pulses for high-resolution NMR. , 2003, Journal of magnetic resonance.

[32]  G. Greuel,et al.  A Singular Introduction to Commutative Algebra , 2002 .

[33]  Jean Charles Faugère,et al.  A new efficient algorithm for computing Gröbner bases without reduction to zero (F5) , 2002, ISSAC '02.

[34]  R. Brockett,et al.  Sub-Riemannian geometry and time optimal control of three spin systems: Quantum gates and coherence transfer , 2001, quant-ph/0106099.

[35]  Dongming Wang,et al.  Elimination Methods , 2001, Texts and Monographs in Symbolic Computation.

[36]  Marc Moreno Maza,et al.  On the Theories of Triangular Sets , 1999, J. Symb. Comput..

[37]  J. Faugère.  A new efficient algorithm for computing Gröbner bases (F4) , 1999 .

[38]  John F. Canny,et al.  An Efficient Algorithm for the Sparse Mixed Resultant , 1993, AAECC.

[39]  Teresa Krick,et al.  An Algorithm for the Computation of the Radical of an Ideal in the Ring of Polynomials , 1991, AAECC.

[40]  B. Bonnard.  Feedback equivalence for nonlinear systems and the time optimal control problem , 1991 .

[41]  John Canny,et al.  The complexity of robot motion planning , 1988 .

[42]  Dima Grigoriev,et al.  Solving Systems of Polynomial Inequalities in Subexponential Time , 1988, J. Symb. Comput..

[43]  D. S. Arnon,et al.  Algorithms in real algebraic geometry , 1988 .

[44]  A. Macovski,et al.  Optimal Control Solutions to the Magnetic Resonance Selective Excitation Problem , 1986, IEEE Transactions on Medical Imaging.

[45]  S. Helgason.  Differential Geometry, Lie Groups, and Symmetric Spaces , 1978 .

[46]  Bruno Buchberger,et al.  A theoretical basis for the reduction of polynomials to canonical forms , 1976, SIGS.

[47]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[48]  M. L. Chambers.  The Mathematical Theory of Optimal Processes , 1965 .

[49]  Lawrence Markus,et al.  VIII. Quadratic Differential Equations and Non-Associative Algebras , 1960 .