Modelling and well-posedness of a nonlinear hybrid system in fed-batch production of 1,3-propanediol with open loop glycerol input and pH logic control

A fed-batch fermentation of glycerol by Klebsiella pneumoniae with open loop substrate input and pH logic control is considered in this paper. We propose a nonlinear hybrid system to describe this process, which consists of discrete variables to represent the flow rate of glycerol and alkali and continuous variables for substance concentrations. The hybrid system includes both time- and state-based switchings, which are respectively determined by a pre-assigned times sequence and an output equation of the pH. It is proved that the hybrid system is non-Zeno. Some basic properties of solutions to the hybrid system are also explored, including existence, uniqueness, boundedness and continuous dependence on initial state and parameters. Additionally, a numerical simulation is carried out to show that the proposed hybrid system can describe the fed-batch culture properly.