Geometric and Numerical Methods in the Contrast Imaging Problem in Nuclear Magnetic Resonance
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Geometric and Numerical Methods in the Contrast Imaging Problem in Nuclear Magnetic Resonance Bernard Bonnard · Mathieu Claeys · Olivier Cots · Pierre Martinon
Received: 26 September 2013 / Accepted: 2 April 2014 © Springer Science+Business Media Dordrecht 2014
Abstract In this article, the contrast imaging problem in nuclear magnetic resonance is modeled as a Mayer problem in optimal control. The optimal solution can be found as an extremal, solution of the Maximum Principle and analyzed with the techniques of geometric control. This leads to a numerical investigation based on so-called indirect methods using the HamPath software. The results are then compared with a direct method implemented within the Bocop toolbox. Finally lmi techniques are used to estimate a global optimum. Keywords Geometric optimal control · Contrast imaging in NMR · Direct method · Shooting and continuation techniques · Moment optimization
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B. Bonnard ( ) Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9 avenue Alain Savary, 21078 Dijon, France e-mail: [email protected] B. Bonnard · O. Cots INRIA Sophia Antipolis Méditerranée, 06902 Sophia Antipolis, France O. Cots e-mail: [email protected] M. Claeys CNRS, LAAS, 7 avenue du colonel Roche, 31077 Toulouse, France e-mail: [email protected] M. Claeys Université de Toulouse, UPS, INSA, INP, ISAE; UT1, UTM, LAAS, 31077 Toulouse, France P. Martinon Inria and Ecole Polytechnique, 91128 Palaiseau, France e-mail: [email protected]
B. Bonnard et al.
1 Introduction The control in Nuclear Magnetic Resonance (NMR) of a spin-1/2 particle is described by the Bloch equation [29] dMx = −Mx /T2 + ωy Mz − ωMy dτ dMy = −My /T2 − ωx Mz + ωMx dτ dMz = (M0 − Mz )/T1 + ωx My − ωy Mx dτ
(1)
where the state variable is the magnetization vector M = (Mx , My , Mz ), the control is the magnetic field ω = (ωx , ωy , 0) bounded by |ω| ≤ ωmax , ωmax = 2π × 32.3 Hz being the experimental intensity of the experiments [26], ω is the resonance offset and τ is the time. In order to normalize our system, we introduce: q = (x, y, z) = (Mx , My , Mz )/M0 and q belongs to the Bloch ball |q| ≤ 1. The time is normalized to t = τ ωmax and we introduce the control u = ω/ωmax , |u| ≤ 1. In this paper, we assume ω = 0, leading to the following normalized system: dx = −Γ x + u2 z dt dy = −Γ y − u1 z dt dz = γ (1 − z) + u1 y − u2 x, dt
(2)
where the parameters Γ = 1/(ωmax T2 ) and γ = 1/(ωmax T1 ) satisfy 2Γ ≥ γ ≥ 0. In the contrast problem, we consider two uncoupled spin-1/2 systems corresponding to different particles, each of them solution of the Bloch equation (2) with respective coefficients (γ1 , Γ1 ), (γ2 , Γ2 ), and controlled by the same magnetic field. By denoting each system by q˙i = f (qi , Λi , u), Λi = (γi , Γi ) and qi = (xi , yi , zi ), this leads to consider the system written shortly as q˙ = F (q, u), where q = (q1 , q2 ) is the state variable and where q˙ means the derivative of q with respect to the time t . In the numerical simulations, we shall consider two
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