L∞-Gain of Fractional-Order Positive Systems With Mixed Time-Varying Delays

This paper investigates the
$L_{\infty }$
-gain of incommensurate fractional-order delayed positive systems (FODPSs), in which a mixture of unbounded delays and distributed delays is considered. Through the utilization of the Banach’s fixed point theorem, the solution to the system is shown to exist uniquely. Then, two necessary and sufficient criteria achieving the positivity and stability of FODPSs with mixed delays are proposed, respectively. For the purpose of calculating the
$L_{\infty }$
-gain, a sample data system is formulated to approximate the lower bound of system trajectories. Additionally, it reveals that the duration of distributed delays have an impact on the
$L_{\infty }$
-gain, whereas unbounded delays will not. Finally, the validity of the theoretical results is explanted through a numerical simulation.