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We present a geometric framework to analyze optimal control problems of uncoupled spin 1/2 particles occurring in nuclear magnetic resonance. According to the Pontryagin's maximum principle, the optimal trajectories are solutions of a pseudo-Hamiltonian system. This computation is completed by sufficient optimality conditions based on the concept of conjugate points related to Lagrangian singularities. This approach is applied to analyze two relevant optimal control issues in NMR: the saturation control problem, that is, the problem of steering in minimum time a single spin 1/2 particle from the equilibrium point to the zero magnetization vector, and the contrast imaging problem. The analysis is completed by numerical computations and experimental results.

The dynamics of a spin 1/2 particle in nuclear magnetic resonance (NMR) is described by the Bloch equation

singular extremals if they satisfy the relation

regular extremals if the control takes its values in the boundary of the control domain.

General extremals are the concatenation of singular and regular arcs. An important example in our study is the case of single-input control systems

Except in some cases, for example, time minimal control for linear systems, the maximum principle is only a necessary optimality condition, and sufficient conditions are related to the concept of extremal field and Hamilton-Jacobi-Bellman equation. This leads to the introduction in optimal control of the difficult notion of conjugate points. Based on recent works [

Time Minimum Problem for a Single Spin. Under suitable coordinates, the system takes the form

Contrast Imaging Problem. In this case, a simplified model is formed by coupling two systems (

We consider a control system of the form

Time Minimum Control Problem. Reach in minimum time from

Mayer Problem. Steer

We denote the accessibility set at time

Let us consider the Hamiltonian function

It is well known that the Pontryagin maximum principle is a set of necessary conditions for our optimal control problem [

Let

The boundary conditions on the adjoint vector are called transversality conditions.

Both optimal problems satisfy the same Hamiltonian dynamics defined by (

The respective boundary conditions define the so-called BC-extremals:

The weak maximum principle consists in replacing the maximality condition (

Relaxing

One then deduces the following results.

If

If

Next, we make computations of extremals which will be used in the sequel.

A system such that the control enters linearly is called affine. We consider a situation for which

The relation

Extremals of order zero correspond to the singularity of the input-output mapping

The system is of the form

Regular extremals: the control is given by

Singular extremals: since the system is linear in

Plugging such

Singular extremals correspond to the singularities of the input-output mapping

From the maximality condition, one has the necessary optimality condition called the Legendre-Clebsch condition

An important relation in our analysis comes from the work of [

For the single-input case, one can define a reduction of the system, using the so-called Goh transformation. Assuming that

For both cases, the extremals are in correspondence, due to the intrinsic interpretation of the singular trajectories as singularities of the input-output mapping. Moreover, introducing the reduced Hamiltonian

We will present the results needed in our analysis to get sufficient second-order optimality conditions under generic assumptions in the singular case. The smooth system is

Our first assumption is the strong-Legendre condition:

The quadratic form

Using the implicit function theorem, the extremal control is then locally defined as a smooth function

Let

The following geometric result is crucial [

Let

This leads to the following definition.

We define the exponential mapping

An algorithm can be deduced.

Let

Seminal works in optimal control are to relate this concept to the optimality properties [

The extremal trajectory

We are in the nonexceptional case where

Under our assumptions, the first geometric conjugate time

Consider the control system

Combined with the CotCot code [

A more complicate and subtle question is to consider the same problem along singular arcs for the single-input affine control system

The concept of conjugate point is associated to optimal control problems with fixed end-points conditions. In the contrast problem, it has to be adapted to the problem with fixed initial condition

For simplicity, we consider here the time-minimal control problem. We pick up a reference extremal trajectory

Excluding

In the next sections, our geometric tools will be applied to our two case studies.

One considers the time minimal control of steering the state

given a point

glue together the local solutions to get the global time optimal solution. More precisely, we will describe the global synthesis to steer

One has

for

for

The interesting case is when

The singular control in the neighborhood of the point

For small time, the optimality status of the singular arc can be deduced from the generalized Legendre-Clebsch conditions. In order to be optimal, we have

hyperbolic case

elliptic case

Applying this test when

the horizontal singular line is a fast direction;

the vertical singular line is fast provided

In particular, this test excludes the standard policy in NMR called the inversion recovery sequence [

In order to complete the optimality analysis, one introduces for 2D-system the following clock-form

Assume

The analysis is completed by the optimality properties of bang-bang extremals. The two main tools are the classification of the regular extremals near the switching surface

The singular extremals are contained in the subset

Hyperbolic Case. at the contact point with

Elliptic Case. at the contact point with

Parabolic Case. Here,

This classification is far from being sufficient to analyze locally our problems. In particular, we have the following situations.

It is a transition between the hyperbolic and the parabolic cases. The singular control

Even more complicated situation can occur, and the most interesting is the one related to the interaction between the two singular lines, which is described next.

Assume that

One calls a bridge between the horizontal and the vertical singular arcs the bang arc such that the concatenation singular-bang-singular arc is optimal.

In our study, the existence of a bridge is related to the following phenomenon. At the saturating point

Next, we present a geometric method to evaluate switching points for 2D-systems. This method is effective if the system is bilinear. The geometric process is described in details in [

Instead of using the adjoint equation to determine the switching sequences, we introduce the following coordinate invariant point of view. Assume 0,

We have by definition

In order to complete the global time minimal synthesis with initial point at the north pole, we must glue together the local syntheses, and the final result is represented in Figure

Global synthesis from the north pole. The switching locus

Note that different works have shown that such optimal control field can be implemented with a very good accuracy in NMR experiments, the standard error being of the order of few percents [

Plot of the optimal trajectories (left) and of the inversion recovery sequence (right) in the plane

For simplicity, we will limit our analysis to the single-input case. The system can be written shortly as

According to the general computations of Section

In order to be optimal, the singular extremals have to satisfy the generalized Legendre-Clebsch condition

The time-optimal solution of the first spin system is solution of the contrast problem provided the transfer time

If the transfer time is not fixed, then according to the maximum principle, this leads to the additional constraint

Meromorphic differential equations of the form (

An important object to analyze our problem is the action of the feedback group

Let

Let

We have the following theorem [

The following diagram is commutative:

Moreover, under mild assumption,

A feedback

The collinear set

It is defined by the constrained Hamiltonian equations given by

This set of equations defines a Hamiltonian vector field on

We introduce the two Poisson brackets:

Geometric invariants are related to the surface

The object of this section is to combine geometric and numerical methods to analyze the contrast problem. The first step is to construct along a reference singular trajectory a

We consider the control system

Under generic assumptions, the first conjugate time

If we apply this result to the contrast problem, one can deduce the simplest result about an extremal policy which provides a local optimal solution. The initial point is the north pole of the Bloch ball, and such a point is singular for the singular flow. Hence, the first control to be applied is

The simplest possible sequence is of the form

More complicated sequences can occur, and from our geometric analysis, this can be related to two phenomena which can be easily computed:

existence of a conjugate time,

saturation of the control and birth of a bridge, which leads to a BSBS policy.

A standard regularization process is the one used in the LQ-control which is an application of Tychonov theorem to control system [

In our nonlinear situation, the convergence is more intricate, but the contrast problem which is a Mayer problem can be interpreted as a cheap control problem. The regularization amounts to the standard homotopy in the cost:

Another regularization process is to use the refined cost

We apply in this section the different tools presented above on a realistic NMR contrast problem corresponding to the cerebrospinal fluid/water case [

We next present some numerical computations. We first analyze the structure of the singular flow. For that purpose, we plot in Figure

Projection of the singular flow onto the planes

It is the same as Figure

It is the same as Figure

Zoom of the results of Figure

In order to improve the contrast, a locally optimal BS-sequence solution of the maximum principle is computed as follows. The transfer time

Evolution of the control field

Trajectories of the magnetization vector for the first and second spins for the control field represented in the bottom panel. This field is the solution of the homotopy problem for

It is the same as Figure

It is the same as Figure

Evolution of the final contrast as a function of the transfer time.

We conclude this paper by illustrating our numerical results with the first results of a contrast experiment. We consider two surfaces as displayed in Figure

Experimental results on the contrast problems. The inner disk mimics the spin 1, while the outside ring mimics the spin 2. The two surfaces are separated by a thin black circle. (a) is a reference image before the application of the control field when the two spins are at the north pole of the Bloch sphere. (b) represent the contrast after a real experiment. The panel (b) corresponds to

The authors acknowledge O. Cots and Y. Zhang for useful discussions and for providing them with some numerical and experimental results of this paper. B. Bonnard and D. Sugny acknowledge support from the PEPS INSIS