针对一类微生物连续发酵生产~$1,3$-丙二醇 ($1,3$-PD)问题,综合考虑胞内、胞外各物质浓度的变化,建立了一个新的非线性系统动力学模型.根据模型中指示函数的不同取值,将连续发酵过程拆分为~16~种切换模式.考虑到部分系统参数依赖于决策变量,讨论了系统状态关于参数与决策变量的灵敏度,得到了含有切换的梯度公式.基于灵敏度函数,建立了切换最优控制模型.应用序列二次规划算法对该模型进行了数值求解,通过大规模计算,得到了甘油流加速率和初始注入浓度的最优策略.数值结果表明,依该流加策略可以有效提高终端时刻~$1,3$-PD~的浓度.

For a class of the microbial continuous fermentation process of glycerol to $1,3$-propanediol ($1,3$-PD), a novel nonlinear dynamical system model is proposed by considering comprehensively the evolution of concentration of the intracellular and extracellular substances. According to the different values of indicator functions in the model, the continuous fermentation process is divided into 16 kinds of switching mode. Considered the parts of the system parameters depend on the decision variables, the sensitivities of the system states with respect to the parameters and the decision variables are discussed, and the gradient formulas containing switches are obtained. A switching optimal control model is formulated based on the sensitivity functions, in which the sequential quadratic programming (SQP) is applied for seeking numerical solution. The optimal feeding rates and the initial poured concentration of glycerol are obtained through large-scale computing. Numerical results show that, by employing the optimal control strategies, the concentration of $1,3$-PD at the terminal time can be increased considerably.