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

2014/07/13 by Gal Hochma, Hochma, Gal, Michael Margaliot +1
Mathematics · #FOS: Mathematics #Optimization and Control (math.OC) #math.OC

paper · pdf · doi:10.48550/arxiv.1407.3437

arxiv created 2014/07/13 · arxiv updated 2014/07/15

Abstract

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.

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