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Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 478081, 7 pages
Research Article

Optimal Control of a Nonlinear Time-Delay System in Batch Fermentation Process

School of Mathematics and Information Science, Shandong Institute of Business and Technology, Yantai 264005, China

Received 24 March 2014; Accepted 30 May 2014; Published 1 July 2014

Academic Editor: Zhan Shu

Copyright © 2014 Yongsheng Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The main control goal in batch process is to get a high yield of products. In this paper, to maximize the yield of 1,3-propanediol (1,3-PD) in bioconversion of glycerol to 1,3-PD, we consider an optimal control problem involving a nonlinear time-delay system. The control variables in this problem include the initial concentrations of biomass and glycerol and the terminal time of the batch process. By a time-scaling transformation, we transcribe the optimal control problem into a new one with fixed terminal time, which yields a new nonlinear system with variable time-delay. The gradients of the cost and constraint functionals with respect to the control variables are derived using the costate method. Then, a gradient-based optimization method is developed to solve the optimal control problem. Numerical results show that the yield of 1,3-PD at the terminal time is increased considerably compared with the experimental data.