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 (Formula presented.) channels with (Formula presented.) 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.