Non-linear eigenvalue problems arising from growth maximization of positive linear dynamical systems

Vincent Calvez (1, 2) , Pierre Gabriel (3) , Stéphane Gaubert (4, 5)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 NUMED - Numerical Medicine
3 LMV - Laboratoire de Mathématiques de Versailles
4 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
5 MAXPLUS - Max-plus algebras and mathematics of decision

We study a growth maximization problem for a continuous time positive linear system with switches. This is motivated by a problem of mathematical biology (modeling growth-fragmentation processes and the PMCA protocol). We show that the growth rate is determined by the non-linear eigenvalue of a max-plus analogue of the Ruelle-Perron-Frobenius operator, or equivalently, by the ergodic constant of a Hamilton-Jacobi (HJ) partial differential equation, the solutions or subsolutions of which yield Barabanov and extremal norms, respectively. We exploit contraction properties of order preserving flows, with respect to Hilbert's projective metric, to show that the non-linear eigenvector of the operator, or the ''weak KAM'' solution of the HJ equation, does exist. Low dimensional examples are presented, showing that the optimal control can lead to a limit cycle.

Mots clés: Limit cycle; Linear differential inclusion; Positive linear system; Optimal control; weak KAM solution; Hamilton-Jacobi-Bellman equation

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Vincent Calvez, Pierre Gabriel, Stéphane Gaubert. Non-linear eigenvalue problems arising from growth maximization of positive linear dynamical systems. IEEE 53rd Annual Conference on Decision and Control (CDC 2014), Dec 2014, Los Angeles, United States. pp.1600--1607, ⟨10.1109/CDC.2014.7039628⟩. ⟨hal-00974749⟩