Journal of Applied Mathematics
Volume 2011 (2011), Article ID 624516, 11 pages
Research Article

Optimal Control for Multistage Nonlinear Dynamic System of Microbial Bioconversion in Batch Culture

1School of Life Science and Biotechnology, Dalian University of Technology, Dalian, Liaoning 116012, China
2School of Mathematical Science, Dalian University of Technology, Dalian, Liaoning 116024, China

Received 30 April 2011; Revised 23 May 2011; Accepted 3 June 2011

Academic Editor: Kuppalapalle Vajravelu

Copyright © 2011 Lei Wang et al. 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.


In batch culture of glycerol biodissimilation to 1,3-propanediol (1,3-PD), the aim of adding glycerol is to obtain as much 1,3-PD as possible. Taking the yield intensity of 1,3-PD as the performance index and the initial concentration of biomass, glycerol, and terminal time as the control vector, we propose an optimal control model subject to a multistage nonlinear dynamical system and constraints of continuous state. A computational approach is constructed to seek the solution of the above model. Firstly, we transform the optimal control problem into the one with fixed terminal time. Secondly, we transcribe the optimal control model into an unconstrained one based on the penalty functions and an extension of the state space. Finally, by approximating the control function with simple functions, we transform the unconstrained optimal control problem into a sequence of nonlinear programming problems, which can be solved using gradient-based optimization techniques. The convergence analysis and optimality function of the algorithm are also investigated. Numerical results show that, by employing the optimal control, the concentration of 1,3-PD at the terminal time can be increased, compared with the previous results.