Periodic solution of a turbidostat system with impulsive state feedback control

A turbidostat is an apparatus used to continuously culturing microorganisms. The concentration of the microorganism in the turbidostat can be controlled by the photoelectricity system or other devices. When the concentration of the microorganism is lower than a critical level, the dilution rate keeps constant. Once the concentration reaches the critical level, the dilution rate can be increased by the control of the photoelectricity system. Based on the design ideas of the turbidostat, a differential equation with impulsive state feedback control, which has no explicit solutions, is proposed for the turbidostat system. By the existence criteria of periodic solution of a general planar impulsive autonomous system, the conditions for the existence of an order one periodic solution of the system are obtained. Furthermore, it is pointed out that the system either tends to a stable state or has a periodic solution. Finally, some discussions and numerical simulations are given.