Abstract
We consider a continuous-time bilinear control system (BCS) with Metzler matrices. Each entry in the transition matrix of such a system is nonnegative, making the positive orthant an invariant set of the dynamics. Motivated by the stability analysis of positive linear switched systems (PLSSs), we define a control as optimal if, for a fixed final time, it maximizes the spectral radius of the transition matrix. Our main result is a first-order necessary condition for optimality in the form of a maximum principle (MP). The proof of this MP combines the standard needle variation with a basic result from the Perron--Frobenius theory of nonnegative matrices. We describe several applications of this MP to the stability analysis of PLSSs under arbitrary switching.
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