### Positivity and Stability of Fractional-Order Linear Time-Delay Systems

HAO Yilin1, HUANG Chengdai2, CAO Jinde3, LIU Heng1

1. 1. School of Mathematics and Physics, Guangxi University for Nationalities, Nanning 530006, China;
2. School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China;
3. School of Mathematics, Southeast University, Nanjing 211189, China
• Received:2021-02-01 Revised:2021-07-24 Online:2022-11-25 Published:2022-12-23
• Contact: LIU Heng,Email:liuheng122@gmail.com
• Supported by:
This work was supported by the National Natural Science Foundation of China under Grant Nos. 61967001, 12062004, and 11771263, the Innovation Project of Guangxi University for Nationalities Graduate Education (gxun-chxps201908 and gxun-chxkc201903), and the Guangxi Natural Science Foundation (2019GXNSFAA185007 and 2020GXNSFAA297240).

HAO Yilin, HUANG Chengdai, CAO Jinde, LIU Heng. Positivity and Stability of Fractional-Order Linear Time-Delay Systems[J]. Journal of Systems Science and Complexity, 2022, 35(6): 2181-2207.

This article focuses on the positivity and the asymptotic stability of fractional-order linear time-delay systems (FOLTDSs) which are composed of $N$ $(N\geq2)$ subsystems. Firstly, a sufficient and necessary condition is given to ensure the positivity of FOLTDSs. The solutions of the studied systems are obtained by using the Laplace transform method, and it can be observed that the positivity of FOLTDSs is completely determined by the series of matrices and independent of the magnitude of time-delays. Secondly, a theorem is given to prove the asymptotic stability of positive FOLTDSs. By considering the monotonicity and asymptotic properties of systems with constant time-delay, it is further shown that the asymptotic stability of positive FOLTDSs is independent of the time-delay. Next, a state-feedback controller, whose gain matrix is derived by resolving a linear programming question, is designed such that the state variables of the systems are nonnegative and asymptotically convergent. When the order of the FOLTDSs is greater than one, by utilizing a proposed property of Caputo derivative, a sufficient condition for the positivity of FOLTDS is presented. Finally, simulation examples are presented to verify the validity and practicability of the theoretical analysis.