Positivity and stability of mixed fractional‐order systems with unbounded delays: Necessary and sufficient conditions

Abstract

This article provides a comprehensive study on quantitative properties of linear mixed fractional-order systems with multiple time-varying delays. The delays can be bounded or unbounded. We first obtain a result on existence and uniqueness of solutions to these systems. Then, we prove a necessary and sufficient condition for their positivity. Finally, we provide a necessary and sufficient criterion to characterize asymptotic stability of positive linear mixed fractional-order systems with multiple time-varying delays.

Impulsive control of unstable homogeneous positive systems of degree one with unbounded time‐varying delay

“…In view of 𝜉 > 0, ( 6), (23), and ( 24), we have ̇x p (t * ) = (6) f p (x(t * )) + g p (x(t * − 𝜏(t * ))) + 𝜉 > (23,24) 0, this contradicts with ̇x p (t * ) ≤ (22) 0, thus (19) holds. By ( 7) and ( 19), we have…”

“…Furthermore, systems with unbounded time‐varying delays are more general but also more challenging to deal with than systems with bounded delays 17 . Therefore, positive systems with unbounded time‐varying delays attract a lot of attention in the past decades, such as $$$ {\ell}_{\infty } $$$ / $$$ {L}_{\infty } $$$‐gain analysis, 18 stability analysis, 19‐22 and stabilization 23 . Feyzmahdavian et al 24 first investigated the asymptotic stability and $$$ \mu $$$‐stability of homogeneous positive systems with unbounded time‐varying delays, which is a special class of nonlinear positive delay systems.…”

Stability Analysis of Finite Time for a Class of Nonlinear Time-Delay Fractional-Order Systems

“…Note that in [11], the authors derived FTS results for time-delay FOS with 1 < σ < 2 using Lemma 1. Furthermore, it is worth highlighting that the authors of [15,16] delved into the exploration of the existence and uniqueness of global solutions, along with the assessment of exponential boundedness, for time-delay FOS in the case where 0 < σ < 1. These investigations were conducted with the aid of Lemma 1.…”