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A family of optimal control problems for a single and two coupled spinningparticles in the Euler-Lagrange formalism is discussed. A characteristic ofsuch problems is that the equations controlling the system are implicit and areduction procedure to deal with them must be carried on.  The reduction of the implicit control equations arising in these problemswill be discussed in the slightly more general setting of implicit equationsdefined by invariant one-forms on Lie groups. As an instance, the first orderdifferential equations describing the extremal solutions of an optimal controlproblem for a single spinning particle, obtained by using Pontryagin's MaximumPrinciple (PMP), will be found and shown to be completely integrable.  Then, using again PMP, solutions for the problem of two coupled spinningparticles will be characterised as solutions of a system of coupled non-linearmatrix differential equations. The reduction of the implicit system will showthat the reduced space for them is the product of the space of states for theindependent systems, implying the absence of `entanglement' in this instance.  Finally it will be shown that, in the case of identical systems, the degreethree matrix polynomial differential equations determined by the optimalfeedback law, constitute a completely integrable Hamiltonian system and some ofits solutions are described explicitly.

Optimal control of two coupled spinning particles in the Euler-Lagrange picture