Modelling and Regularity of Nonlinear Impulsive Switching Dynamical System in Fed‐Batch Culture

Abstract

A hybrid system with state‐based switchings is proposed to describe the fed‐batch production of 1,3‐propandiol from glycerol in our previous work. However, the on‐off switching of alkali is too frequent, which greatly increases the computational cost of the numerical solution to the system so as to locate the state‐based switchings in strict time order and implement the correct mode changes. To deal with this problem, we consider the switching of alkali pump as an impulsive event and present a nonlinear impulsive switching system to describe the fed‐batch culture. It is proved that the impulsive switching system is non‐Zeno. Some basic properties of solutions to the impulsive switching system are also explored. In order to overcome the discontinuities of the system, the Skorohod topology is induced and a specific form of λ is constructed to prove the main theorem. Additionally, a numerical simulation is carried out to show that the proposed system can describe the fed‐batch culture properly and the essential difference with the previous work.