Stability And Optimal Control Of Nonlinear System In Microorganism Fermentation

Based on the practical background of producing 1,3-propancdiol (1,3-PD) by micro-bial continuous and batch fermentation, this dissertation studies stability and optimal control of nonlinear dynamic system. The dissertation is part of the National Natural Science Foundation of China " Optimality theory and algorithm in nonlinear piccewise smooth dynamic system " (Grant No.10471014), the tenth five-year plan of science and technology of China " Microbial production of 1,3-propancdiol " (Project No. 2001BA708B01-04), and Lifc+X cross-subject plan in Dalian University of Technology" Study on optimal control in the process of microbial production of 1,3-propancdiol ". The contents of the dissertation include the existence of equilibrium and the stability criterion, and that a nonlinear optimal problem with equality and inequality constraint is concluded taking asymptotic stable equilibrium in continuous system as main constraint condition. At the same time, two other optimal control models are formed taking nonlinear dynamic system in continuous and batch culture as state equations separately. The existence of optimal solution, optimality conditions, optimality functions, and optimal algorithms are studied in every model.The main contributions arc as follows:1. The stability of equilibrium of non-linear dynamic system for microorganism in continuous culture is considered first. The existence of equilibrium is proved, the conclusion that the equilibrium is the continuous function of the dilution rate and substrate concentration in medium in a certain range is concluded, and the stability criterion of equilibrium is obtained. In theory, it explains the phenomena of multiplicity shown in laboratory. It means, in order that the concentration of 1,3-PD attains the expected value when the system approach to the stable equilibrium state, we can choose the proper dilution rate and substrate concentration in medium in practice.2. The optimal control problems for productivity of 1,3-PD from continuous and batch fermentation of microorganism arc considered. Taking the maximum production strength of 1,3-PD as the objective function, the nonlinear dynamical system of continuous and batch culture as main constraint condition respectively, two optimal control models are given. In every model, the system and its solution's property are analyzed, the existence of optimal solution and one-order necessary optimality condition arc discussed according the optimal theory and methods of non-differcntiable function. Two optimality functions for above optimal control models arc introduced, and the cquiva-lence between the optimality function and the one-order necessary optimality condition is concluded. It shows, in theory, by the optimal control model, we can find the optimal operating conditions under which the production strength of 1,3-PD will attain the maximum.3. In the process of continuous culture, how to get the highest concentration of 1,3-PD when the system attains the equilibrium state is the aim of producers. This actual problem is considered as a nonlinear optimal problem with equality and inequality constraint. The existence of optimal solution and the optimality condition are analyzed. The optimality function of the nonlinear optimal problem is defined, and the equivalence between the zero of optimality function and necessary optimality condition is proved by infinite-dimensional optimization theory.4. For above nonlinear optimal control models and nonlinear optimal problem, corresponding algorithms arc formed. First, taking the optimality function as the terminal criteria, an algorithm is given for the nonlinear optimal problem that takes the asymptotic stable equilibrium as main constraint condition in continuous culture. The convergence; of the algorithm is proved. Next, for the optimal control problem in continuous culture, a discrete optimal control problem is followed by the discrete dynamics. The necessary optimality condition is concluded. The optimality function that is consistent approximation to the optimality function of the optimal control problem is defined in discrete optimal control problem. Taking its optimality function as terminal criteria, an optimal algorithm for the discrete optimal control problem is formed, and a recapitulative algorithm for the optimal control problem in continuous culture is constructed too. Its convergence is proved. At last the optimal control problem in batch culture is analyzed by similar method and the result is followed.