We analyse an optimal control with the following features: the dynamical
system is linear, and the dependence upon the control parameter is affine. More
precisely we consider x˙α(t)=(G+α(t)F)xα(t), where
G and F are 3×3 matrices with some prescribed structure. In the
case of constant control α(t)≡α, we show the existence of an
optimal Perron eigenvalue with respect to varying α under some
assumptions. Next we investigate the Floquet eigenvalue problem associated to
time-periodic controls α(t). Finally we prove the existence of an
eigenvalue (in the generalized sense) for the optimal control problem. The
proof is based on the results by [Arisawa 1998, Ann. Institut Henri Poincar\'e]
concerning the ergodic problem for Hamilton-Jacobi equations. We discuss the
relations between the three eigenvalues. Surprisingly enough, the three
eigenvalues appear to be numerically the same