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