New asymptotic stability theory for real order systems and applications

Abstract

Metzler asymptotic stability of real order systems gives hidden paths of nontrivial solutions to geometric constant solutions as time tends to ∞ in positive and negative orthants of Euclidean space when associated with random initial-time. Suppose that there is a governing law that describes the flow of geometric variables described by a real-order system in Euclidean space and a geometric solution zero vector that satisfies such a system. We introduce the concepts of orthant Metzler asymptotic stability that give rise to the notion of so-called new L-asymptotic stability. We prove that there exist order-dependent conditions that give a new L-asymptotic stability. The main results of this paper are comparison theorems that give L-asymptotic stability, which is essential in finding convergence of non-negative and non-positive solutions to geometric constant solutions. We establish upper and lower theoretical α-order Mittag-Leffler bounds of the Euclidean norm measure of non-trivial solutions to such systems, which characterize the Mittag-Leffler rate of decay. As applications of some theoretical results, we provide three examples that include a feedback method to control cancer in a cancer chemotherapy model to illustrate the effectiveness.