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 irreducible matrices. Then to overcome the unsolvability of the optimal control, we relax the problem. In the two dimensional case, we solve explicitly the optimal relaxed control problem when the final time is large enough.