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

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 ofλis 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.

Point processes and random measures

The purpose of this paper is to give a short introduction to the theory of point processes and random measures. Our hope is that the paper can be used as a complement to a study of Billingsley's book Convergence of Probability Measures. The subjects treated are: topological properties of the space of realizations; characterization and convergence of random measures with special attention to simple point processses and diffuse random measures; the relation between the topology used and the Skorohod topology. Some thinning and superposition results are given without proofs.

Point processes and random measures

The purpose of this paper is to give a short introduction to the theory of point processes and random measures. Our hope is that the paper can be used as a complement to a study of Billingsley's book Convergence of Probability Measures. The subjects treated are: topological properties of the space of realizations; characterization and convergence of random measures with special attention to simple point processses and diffuse random measures; the relation between the topology used and the Skorohod topology. Some thinning and superposition results are given without proofs.