  We investigate an optimal control problem which arises in the optimization of
an amplification technique for misfolded proteins. The improvement of this
technique may play a role in the detection of prion diseases. The model
consists in a linear system of differential equations with a nonlinear control.
The appearance of oscillations in the numerical simulations is understood by
using the Perron and Floquet eigenvalue theory for nonnegative irreductible
matrices. Then to overcome the unsolvability of the optimal control, we relax
the problem. In the two dimensional case we solve explicitely the optimal
relaxed control problem when the final time is large enough.
