Nonlinear dynamical systems of bio-dissimilation of glycerol to 1,3-propanediol and their optimal controls

In this study simplified mathematical models of the nonlinear dynamic systems of dissimilation of glycerol to 1,3-propanediol by {\it Klebsiella pneumoniae} in continuous, batch and fed-batch cultures are investigated. Considering big errors between the experimental results and computational values in the existing models, the parameter identification models for these systems are established. The properties of the solutions for the nonlinear dynamic systems are discussed and the identifiability of the parameters is proved. In view of the sudden increase of the glycerol and alkali in fed-batch culture, this paper proposes a nonlinear impulsive system of fed-batch culture. The existence, uniqueness and regularity properties of piecewise solution for the system are proved. Based on the nonlinear impulsive system, the paper constructs an optimal control model in view of the controllability of volumes of glycerol added to the reactor instantaneously, and the existence of the optimal control is obtained.