Extension of the Perron-Frobenius theorem: from linear to homogeneous

This paper deals with homogeneous cooperative systems, a class of positive systems. It is shown that they admit a fairly simple asymptotic behavior, thereby generalizing the well-known Perron-Frobenius theorem from linear to homogeneous systems. As a corollary a simple criterion for global asymptotic stability is established. Then these systems are subject to constant inputs and we prove that asymptotic stability of the uncontrolled system is inherited by the new equilibrium point of the controlled system. Recent results on monotone control systems indicate the importance of this property in proving small gain theorems.