Application of the Pontryagin maximum principle to the time-optimal control in a chain of three spins with unequal couplings

We solve a time-optimal control problem in a linear chain of three coupled spins 1/2 with unequal couplings. We apply the Pontryagin maximum principle and show that the associated Hamiltonian system is the one of a three-dimensional rigid body. We express the optimal control fields in terms of the components of the classical angular momentum of the rigid body. The optimal trajectories and the minimum control time are given in terms of elliptic functions and elliptic integrals.

Figure 1

Panels (a) and (b) correspond to the oscillating and rotating cases, respectively. The light and dark colored surfaces represent $S_1$ and $S_2$, respectively. The optimal trajectories are the intersection between these two surfaces. Rotating solutions occur when the radius $\ell$ of the sphere is larger than the maximum radius of the crushed cylinder (here 1). Panel (c) depicts the extremals for different level sets $\ell$ of the angular momentum (where the sphere has been removed for clarity). We recognize the pendulum phase portrait projected onto $S_1$. Separatrices are plotted in dotted lines.

Figure 2

Definition of the Euler angles.

Figure 3

Plot of the extremals described by the three axes of $R\left(t\right)$ on the unit sphere for the transfer from $(1,0,0)$ to $(0,0,1)$ for $k=2$. The black solid line shows the trajectory drawn by $\vec{r}\left(t\right)$. The blue dotted line corresponds to $\vec{p}/\ell$, and the red dashed one shows $\vec{L}/\ell$.

Figure 4

(a) $f_k\left(\ell \right)$ plotted for $k\in \{1,1.2,2,10\}$. The value of $\ell$ is the solution of the equation $f_k\left(\ell \right)=0$. (b) Corresponding trajectories $(r_1\left(t\right),r_2\left(t\right),r_3\left(t\right))$ on the sphere. The line styles are equal in both panels. The extremals for $k=1$ and $k=1.1$ are very close to each other. (c) Solution $\ell$ of $f_k\left(\ell \right)=0$ in terms of $k$, where a logarithmic scale is used for the $x$ axis. (d) Minimum time $T=t_f$ as a function of $k$. All variables are unitless.

Figure 5

Plot of Jacobi elliptic functions $\mathrm{sn}\left(u\right|m)$ (solid line), $\mathrm{cn}\left(u\right|m)$ (dotted line), and $\mathrm{dn}\left(u\right|m)$ (dashed-dotted line) for $m=1/2$.