High-order maximum principles for the stability analysis of positive bilinear control systems

We consider a continuous-time positive bilinear control system, which is a bilinear control system with Metzler matrices. The positive orthant is an invariant set of such a system, and the corresponding transition matrix is entrywise nonnegative for all time. Motivated by the stability analysis of positive linear switched systems under arbitrary switching laws, we define a control as optimal if it maximizes the spectral radius of the transition matrix at a given final time. We derive high-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 a simple eigenvalue with the generalized Legendre-Clebsch condition and the Agrachev–Gamkrelidze second-order optimality condition.