On Positive Linear Volterra-Stieltjes Differential Systems

We first introduce the notion of positive linear Volterra-Stieltjes differential systems. Then, we give some characterizations of positive systems. An explicit criterion and a Perron-Frobenius type theorem for positive linear Volterra-Stieltjes differential systems are given. Next, we offer a new criterion for uniformly asymptotic stability of positive systems. Finally, we study stability radii of positive linear Volterra-Stieltjes differential systems. It is proved that complex, real and positive stability radius of positive linear Volterra-Stieltjes differential systems under structured perturbations coincide and can be computed by an explicit formula. The obtained results in this paper include ones established recently for positive linear Volterra integro-differential systems [36] and for positive linear functional differential systems [32]-[35] as particular cases. Moreover, to the best of our knowledge, most of them are new. The first author is supported by the Alexander von Humboldt Foundation.