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.