  We consider a continuous-time positive bilinear control system (PBCS), i.e. a
bilinear control system with Metzler matrices. The positive orthant is an
invariant set of such a system, and the corresponding transition matrix C(t) is
entrywise nonnegative for all time t>0. Motivated by the stability analysis of
positive linear switched systems (PLSSs) under arbitrary switching laws, we fix
a final time T>0 and define a control as optimal if it maximizes the spectral
radius of C(T). A recent paper developed a first-order necessary condition for
optimality in the form of a maximum principle (MP). In this paper, we derive
higher-order necessary conditions for optimality for both singular and
bang-bang controls. Our approach is based on combining results on the
second-order derivative of the spectral radius of a nonnegative matrix with the
generalized Legendre-Clebsch condition and the Agrachev-Gamkrelidze
second-order optimality condition.
