[1]
|
E. Allgower and K. Georg, Introduction to Numerical Continuation Methods, vol. 45 of Classics in Applied Mathematics, Soc. for Industrial and Applied Math., Philadelphia, PA, USA, 2003.
doi: 10.1137/1.9780898719154.
|
[2]
|
J. T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2001.
|
[3]
|
F. J. Bonnans, P. Martinon and V. Grélard, Bocop - A Collection of Examples, Technical report, INRIA (2012) RR-8053.
|
[4]
|
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.
|
[5]
|
B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, vol 40 of Mathematics and Applications, Springer-Verlag, Berlin, 2003.
|
[6]
|
B. Bonnard, M. Chyba, A. Jacquemard and J. Marriott, Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance. Mathematical Control and Related Fields-AIMS, Special issue in the honor of Bernard Bonnard. Part II., 3 (2013), 397-432.
doi: 10.3934/mcrf.2013.3.397.
|
[7]
|
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.
|
[8]
|
B. Bonnard, M. Claeys, O. Cots and P. Martinon, Geometric and numerical methods in the contrast imaging problem in nuclear magnetic resonance, Acta Appl. Math., 135 (2015), 5-45.
doi: 10.1007/s10440-014-9947-3.
|
[9]
|
B. Bonnard and O. Cots, Geometric numerical methods and results in the control imaging problem in nuclear magnetic resonance, Math. Models Methods Appl. Sci., 24 (2014), 187-212.
doi: 10.1142/S0218202513500504.
|
[10]
|
B. Bonnard, O. Cots, S. 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.
|
[11]
|
B. Bonnard, O. Cots, J. Rouot and T. Verron, Working Notes on the Time Minimal Saturation of a Pair of Spins and Application in Magnetic Resonance Imaging, http://hal.archives-ouvertes.fr/hal-01721845/
|
[12]
|
B. Bonnard, J.-C. Faugére, A. Jacquemard, M. Safey El Din and T. Verron, Determinantal sets, singularities and application to optimal control in medical imagery, Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation, 103–110, ACM, New York, 2016.
|
[13]
|
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.
|
[14]
|
U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Springer SMAI, 2004.
|
[15]
|
R. Bulirsch and J. Stoer, Introduction to Numerical Analysis, vol.12 of Texts in Applied Mathematics, Springer-Verlag, New York, 2$^{nd}$ edition, 1993.
doi: 10.1007/978-1-4757-2272-7.
|
[16]
|
J.-B. Caillau, O. Cots and J. Gergaud, Differential continuation for regular optimal control problems, Optimization Methods and Software, 27 (2011), 177-196.
doi: 10.1080/10556788.2011.593625.
|
[17]
|
M. Claeys, J. Daafouz and D. Henrion, Modal occupation measures and {LMI} relaxations for nonlinear switched systems control, Automatica, 64 (2016), 143-154.
doi: 10.1016/j.automatica.2015.11.003.
|
[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.
|
[19]
|
M. Gerdts, Optimal Control of ODEs and DAEs, ed. De Gruyter, Berlin, 2012.
doi: 10.1515/9783110249996.
|
[20]
|
D. Henrion, J. B. Lasserre and J. Löfberg, GloptiPoly 3: Moments, optimization and semidefinite programming, Optim. Methods and Software, 24 (2009), 761-779.
doi: 10.1080/10556780802699201.
|
[21]
|
A. J. Krener, The high order maximal principle and its application to singular extremals, SIAM J. Control Optim., 15 (1977), 256-293.
doi: 10.1137/0315019.
|
[22]
|
I. Kupka, Geometric theory of extremals in optimal control problems. Ⅰ. The fold and Maxwell case, Trans. Amer. Math. Soc., 299 (1987), 225-243.
doi: 10.2307/2000491.
|
[23]
|
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.
|
[24]
|
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., 589 (2012).
|
[25]
|
J.-B. Lasserre, Moments, Positive Polynomials and Their Applications, Imperial College Press, London, 2010.
|
[26]
|
J.-B. Lasserre, D. Henrion, C. Prieur and E. Trélat, Nonlinear optimal control via occupation measures and LMI-relaxations, SIAM J. Control Optim., 47 (2008), 1643-1666.
doi: 10.1137/070685051.
|
[27]
|
M. H. Levitt, Spin Dynamics: Basics of Nuclear Magnetic Resonance, John Wiley and Sons, New York-London-Sydney, 2008.
|
[28]
|
L. Markus, Quadratic differential equations and non-associative algebras, Princeton Univ. Press, Princeton, N.J., 5 (1960), 185-213.
|
[29]
|
H. Maurer, Numerical solution of singular control problems using multiple shooting techniques, J. Optim. Theory Appl., 18 (1976), 235-257.
doi: 10.1007/BF00935706.
|
[30]
|
A. Mosek, The Mosek Optimization Toolbox for Matlab Manual, Version 7.1 Revision, 2015.
|
[31]
|
J. Nocedal and S. J. Wright, Numerical Optimization, Springer-Verlag, New York, 1999.
doi: 10.1007/b98874.
|
[32]
|
L. S. Pontryagin, V. G. Boltyanskiĭ, 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 and Sons, Inc., New York-London, 1962.
|
[33]
|
M. Putinar, Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J., 42 (1993), 969-984.
doi: 10.1512/iumj.1993.42.42045.
|
[34]
|
H. Schättler, The local structure of time-optimal trajectories in dimension three under generic conditions, SIAM J. Contr. Opt., 26 (1988), 899-918.
doi: 10.1137/0326050.
|
[35]
|
H. J. Sussmann, Time-optimal control in the plane,, in Feedback Control of Linear and Nonlinear Systems, (Bielefeld/Rome, 1981), vol. 39 of Lecture Notes in Control and Inform. Sci., Springer, Berlin, (1982), 244–260.
doi: 10.1007/BFb0006833.
|
[36]
|
H. J. Sussmann, Regular synthesis for time-optimal control of single-input real-analytic systems in the plane, SIAM J. Control and Opt., 25 (1987), 1145-1162.
doi: 10.1137/0325062.
|
[37]
|
E. Van Reeth, H. Ratiney, M. Tesch, D. Grenier, O. Beuf, S. J. Glaser and D. Sugny, Optimal control design of preparation pulses for contrast optimization in MRI, J. Magn. Reson., 279 (2017), 39-50.
|