Experimental design suboptimization for the enzyme-catalytic nonlinear time-delay system in microbial batch culture

In this paper, an enzyme-catalytic nonlinear time-delay system is investigated to describe the batch culture of glycerol to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae (K. pneumoniae), in which glycerol and 1,3-PD are assumed to pass the cell membrane by passive diffusion coupled with facilitated transport. The existence, uniqueness, boundness of solutions of the system and continuity of solutions with respect to the uncertain parameter appeared in the studied system are also discussed. The purpose of this article is to maximize the productivity of 1,3-PD at terminal time by controlling the initial value of the state vector and terminal time. With this in mind, taking the productivity of 1,3-PD at terminal time as the cost function and the initial concentration of biomass, glycerol, and terminal time as the control vectors, we firstly propose an optimal control model subject to the time-delay system and continuous state inequality constraints. To seek the optimal productivity of 1,3-PD at terminal time and the optimal control, a modified Nelder–Mead algorithm involved with the constraint transcription and smoothing techniques is then developed. Finally, the numerical results illustrate the appropriateness of the enzyme-catalytic nonlinear dynamic system and the validity of the optimization algorithm.