  We study a growth maximization problem for a continuous time positive linear
system with switches. This is motivated by a problem of mathematical biology
(modeling growth-fragmentation processes and the PMCA protocol). We show that
the growth rate is determined by the non-linear eigenvalue of a max-plus
analogue of the Ruelle-Perron-Frobenius operator, or equivalently, by the
ergodic constant of a Hamilton-Jacobi (HJ) partial differential equation, the
solutions or subsolutions of which yield Barabanov and extremal norms,
respectively. We exploit contraction properties of order preserving flows, with
respect to Hilbert's projective metric, to show that the non-linear eigenvector
of the operator, or the "weak KAM" solution of the HJ equation, does exist. Low
dimensional examples are presented, showing that the optimal control can lead
to a limit cycle.
