The PDP feedback control design and exponential stability for a star-shaped network of open channels

This paper investigates the stability of a star-shaped network of open channels under the position and delayed position (PDP) feedback controller. Firstly, we deal with the special case of a single channel, and the well-posedness and exponential stability of the closed-loop system are established by using the semi-group theory and a strict Lyapunov function. The explicit dissipative conditions for system parameters and time-delay are presented. Then, we analyze the delay-independent stability of the closed-loop system by the method of characteristic. Next, we extend the PDP control design to the star-shaped network of n channels with n - 1 inlet channels and one outlet channel, and the exponential stability can be obtained similarly. Finally, we give some numerical simulations for the special case of three channels.