# American Institute of Mathematical Sciences

2015, 5(4): 369-380. doi: 10.3934/naco.2015.5.369

## Nonlinear state-dependent impulsive system in fed-batch culture and its optimal control

 1 School of Mathematical Science, Huaiyin Normal University, No.111, Changjiang West Road, Huai'an 223300, China, China 2 School of Mathematics and Information Science, Shandong Institute of Business and Technology, Yantai, 264005

Received  January 2015 Revised  October 2015 Published  October 2015

In fed-batch culture, feeding substrates is to provide sufficient nutrition and reduce inhibitions simultaneously for cells growth. Hence, when and how much to feed substrates are important during the process. In this paper, a nonlinear impulsive controlls system, in which the volume of feeding is taken as the control function, is proposed to formulate the fed-batch fermentation process.In the system, both impulsive moments and jumps size of state are state-dependent. Some important properties of the system are investigated. To maximize the concentration of target product at the terminal time, an optimal control model involving the nonlinear state-dependent impulsive controlled system is presented.The optimal control problem is subject to the continuous state inequality constraint and the control constraint. The existence of optimal control is also obtained. In order to derive the optimality conditions, the optimal control model is transcribed into an equivalent one by treating the constraints. Finally, the optimality conditions of the optimal control model are obtained via calculus of variations.
Citation: Bangyu Shen, Xiaojing Wang, Chongyang Liu. Nonlinear state-dependent impulsive system in fed-batch culture and its optimal control. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 369-380. doi: 10.3934/naco.2015.5.369
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##### References:
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