Time minimal saturation of a pair of spins and application in Magnetic Resonance Imaging

Figure 1.  In this example, $(\gamma, \Gamma) = (0.12, 0.5)$. (Left) Collinearity set, singular set and visualization of $N$, $O$, $O_1$, $S_1$, $S_1'$, $S_3$ and $S_3'$. (Right) Sign of $\alpha(q)$ in the domain $y\le 0$. The sign in the domain $y\ge 0$ is given by symmetry

Figure 2.  The points $S_1$, $S_2$ and $S_3$ with different orders. Left sub-graph: $(\gamma,\Gamma) = (0.1, 0.5)$ and right: $(\gamma,\Gamma) = (0.163,0.5)$

Figure 6.  Schematic time minimal synthesis to steer a single spin system from the North Pole N to any point of the Bloch ball in the reachable set, for parameters $(\Gamma,\gamma) \in C$. An arbitrary zoom has been used to construct the figure. The set of $\Sigma_i$ forms the switching surface $\Sigma$ dividing the $+1$ and $-1$ areas respectively in red and blue. The minimal time trajectory to steer the spin from N to O is $NS_1S_2S_2'O$, i.e. it is of the form $\sigma _+^N \sigma _s^h \sigma _+^b \sigma _s^v$ with horizontal $\sigma _s^h$ and vertical $\sigma _s^v$ singular arcs. The spin leaves the horizontal singular arc before the point $S_3$ (where the control saturates the constraint) producing a bridge $\sigma _+^b$ to reach the vertical singular line

Figure 7.  Schematic time minimal synthesis to steer a single spin system from the North Pole N to any point of the Bloch ball in the reachable set, for parameters $(\Gamma,\gamma) \in B$. An arbitrary zoom has been used to construct the figure. The set of $\Sigma_i$ forms the switching surface $\Sigma$ dividing the $+1$ and $-1$ areas respectively in red and blue. The minimal time trajectory to steer the spin from N to O is $NS_1'O$, i.e. it is of the form $\sigma _+^N \sigma _s^v$

Figure 8.  Schematic time minimal synthesis to steer a single spin system from the North Pole N to any point of the Bloch ball in the reachable set, for parameters $(\Gamma,\gamma) \in A_2$. For $(\Gamma,\gamma) \in A_1$, the synthesis is the same but there is no horizontal singular line inside the Bloch ball. An arbitrary zoom has been used to construct the figure. The minimal time trajectory to steer the spin from N to O is $NS_1'O$, i.e. it is of the form $\sigma _+^N \sigma _s^v$

Figure 3.  The domain of interest of the parameters in white with the sub-domains $A_1$, $A_2$, $B$ and $C$

Figure 4.  The sub-domains in $(\Gamma,\gamma/\Gamma)$ coordinates. One can notice that when $\Gamma$ tends to $0$, then the slopes of the curves $S_1 = S_2$ and $S_1 = S_3$, in $(\Gamma,\gamma)$ coordinates, are equal to $2/3$

Figure 5.  The slopes $T_2/T_1 = \gamma/\Gamma$ for the species from Table 1 with the particular case when $\omega_\mathrm{max} = 2 \pi\times 32.3$ Hz. This case is represented by a bullet while the slope is represented by a line

Figure 14.  Water case: $\rho = \bar{\rho}$, $\theta = 0.7854$ and $\varepsilon = 0.1$. Trajectories of spins 1 and 2, and control

Figure 9.  Deoxygenated blood case ($C_1$) with RF-inhomogeneity ($\varepsilon = 0.1$). Trajectories for spin 1 and 2 in the (y, z)-plane are portrayed in the first two subgraphs. The corresponding control is drawn in the right subgraph. The horizontal lines $z_1 = z_2 = z_s = \gamma/(2\delta)$, $\delta = \gamma-\Gamma$, is represented by dashed lines. Note that the last bang arc is not well captured by the direct solver because it is too short

Figure 10.  Water case ($C_4$) with RF-inhomogeneity ($\varepsilon = 0.1$). Trajectories for spin 1 and 2 in the (y, z)-plane are portrayed in the first two subgraphs

Figure 11.  Saturation problem of one spin and two spins for the cases $C_1$, $C_2$, $C_3$ and $C_4$. Relative error $\mathrm{err}(r) = (t_f-T_{LMI'}^r)/t_f$ where $r$ is the order of relaxation, $T_{LMI'}^r$ is the optimal value of (59) using the formulation (57) and $t_f$ is the final time computed with $\texttt{Bocop}$

Figure 12.  The relaxation parameters are given by $\rho = \bar{\rho}$ and (Left) $\theta = \theta_\mathrm{1a,1}^+$, (Right) $\theta = \theta_\mathrm{1a,1}^-$. Each subgraph represents the graph of the switching function $H_1$ along the extremal. For $\theta = \theta_\mathrm{1a,1}^-$, we observe that $H_1$ crosses $0$ twice. A singular arc must be added to continue the homotopy on $\theta$. A zoom in on the graph of $H_1$ is given and may be located thanks to the vertical red lines. Note that we add a singular arc, since in this case the singular extremal is time-minimizing for small time

Figure 13.  The homotopies H1a, H1b, H2 and H3 are presented respectively in blue, red, black and black. The homotopies H2 and H3 are presented only on the top-right subgraph. The plain and dashed lines distinguish the different structures. The top-left subgraph gives the cost (i.e. the final time $t_f$) with respect to $\theta$ for the homotopies H1a and H1b. One can see (it is more visible in the zoom) the intersection of the two paths of zeros in terms of cost. H1b is better for small value of $\theta$ and H1a is better for greater values. One can notice that this two paths of zeros are distinct from the bottom-left subgraph. This subgraph gives the norm of the initial adjoint vector with respect to the homotopic parameter. The norm of the shooting function along the paths H1a and H1b is given in the bottom-left subgraph.% One can see the very good accuracy. The strategies from H1a and H1b are compared with homotopies H2 and H3 on the top-right subgraph

Figure 15.  Fat case: $\rho = \bar{\rho}$, $\theta = 0.4636$ and $\varepsilon = 0.1$. Trajectories of spins 1 and 2, and control

Figure 16.  Fluid case: $\rho = \bar{\rho}$, $\theta = 0.1489$ and $\varepsilon = 0.1$. Trajectories of spins 1 and 2, and control

Figure 17.  H2: $\rho = \bar{\rho}$, $\theta = 0.2$ and $\varepsilon = 0.1$. Trajectories of spins 1 and 2, control and $H_1$