New insight to Metzler asymptotic attractive in fractional order systems

Extending beyond the notion of Metzler’s asymptotic stability of equilibria in fractional order systems remains a very challenging and long-standing open problem in the literature. Suppose you are given a fixed bounded solution or any solution pair to such fractional order systems. Is it possible to predict the Metzler asymptotic attractive of a fixed solution or any solution pair of such systems? We introduce a new concept, LU-asymptotic attractive, that holds key concepts to both non-negative and non-positive Metzler asymptotic attractive of fixed solution or any solution pair. We establish some typical new knowledge of order-dependent theoretical results that identifies constant Metzler and non-negative matrices and gives attractive notions to considered solutions. Whenever the constant Metzler matrix was lower and upper majorized by scaling constants, it gives rise to multi-order Mittag-Leffler decay. We have shown that some theoretical results are applicable to practical models arising in the literature.