This paper considers the
problem of steering control via an input
electro-magnetic field for a
system of two interacting spin
$\frac{1}{2}$ particles. The describing model is a bilinear system
whose state varies on the Lie group of special unitary matrices of
dimension $4$, $SU(4)$. By using decompositions of the latter Lie group,
the problem can be decomposed into a number of subproblems for a
system whose state varies on the (smaller) Lie group of $4 \times 4$
proper orthogonal matrices, $SO(4)$. We tackle the time optimal control
problem with a bound on the magnitude of the control field
for this system and show that the optimal minimum time controls can be computed
explicitly and they are the superposition of a constant field and a
sinusoidal one.
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