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## A linear optimal feedback control for producing 1, 3-propanediol via microbial fermentation

 1 School of Maritime Economics and Management, Dalian Maritime University, Dalian 116026, China 2 Collaborative Innovation Center for Transport Studies@Dalian Maritime University, Dalian 116026, China 3 School of Mathematical Science, Dalian University of Technology, Dalian, Liaoning 116024, China 4 Department of Mathematics, Loyola Marymount University, Los Angeles CA 90045, USA

* Corresponding author: Lei Wang

Received  January 2018 Revised  April 2018 Published  September 2019

Fund Project: This work was supported by the National Natural Science Foundation of China(Grants Nos.11371164, 11771008 and 61473326), the National Natural Science Foundation for the Youth of China(Grants Nos.11401073 and 11501574), the Fundamental Research Funds for Central Universities in China(Grants No. DUT19LK37), and the Natural Science Foundation of Shandong Province in China(Grants Nos. ZR2015FM014, ZR2015AL010 and ZR2017MA005).

In this paper, we consider a multistage feedback control strategy for the production of 1, 3-propanediol(1, 3-PD) in microbial fermentation. The feedback control strategy is widely used in industry, and to the best of our knowledge, this is the first time it is applied to 1, 3-PD. The feedback control law is assumed to be linear of the concentrations of biomass and glycerol, and the coefficients in the controller are continuous. A multistage feedback control law is obtained by using the control parameterization method on the coefficient functions. Then, the optimal control problem can be transformed into an optimal parameter selection problem. The time horizon is partitioned adaptively. The corresponding gradients are derived, and finally, our numerical results indicate that the strategy is flexible and efficient.

Citation: Honghan Bei, Lei Wang, Yanping Ma, Jing Sun, Liwei Zhang. A linear optimal feedback control for producing 1, 3-propanediol via microbial fermentation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020095
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##### References:
The evolution of the time grids in 100 hours
The changes of control parameters, $\xi_1(t)$ and $\xi_2(t)$, over time
The change of concentration of biomass, glycerol, 1, 3-PD, acetate and ethanol in continuous culture process in 100 hours
The values of some parameters used in Eqs. (1) - (4)
 $i$ $m_i$ $Y_i$ $\Delta{q_i}$ $k_i$ $b_i$ $c_i$ 1 - - - - 0.025 0.06 2 2.20 0.0082 28.58 11.43 5.18 50.45 3 -2.69 67.69 26.59 15.50 - - 4 -0.97 33.07 5.74 85.71 - -
 $i$ $m_i$ $Y_i$ $\Delta{q_i}$ $k_i$ $b_i$ $c_i$ 1 - - - - 0.025 0.06 2 2.20 0.0082 28.58 11.43 5.18 50.45 3 -2.69 67.69 26.59 15.50 - - 4 -0.97 33.07 5.74 85.71 - -
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