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 and Optimization, 2015, 5 (4) : 369-380. doi: 10.3934/naco.2015.5.369
References:
[1]

J. Angelova and A. Dishliev, Optimization problems for one-impulsive models from population dynamics, Nonlinear Anal., 39 (2000) 483-497. doi: 10.3934/dcdsb.2004.4.1065.

[2]

S. A. Attia, V. Azhmyakov and J. Raisch, On an optimization problem for a class of impulsive hybrid systems, Discrete Event. Dyn. Syst., 20 (2010), 215-231. doi: 10.3934/dcdsb.2004.4.1065.

[3]

V. Azhmyakov, V. G. Boltyanski and A. Poznyak, Optimal control of impulsive hybrid systems, Nonlinear Anal. Hyb. Syst., 2 (2008), 1089-1097. doi: 10.3934/dcdsb.2004.4.1065.

[4]

H. Biebl, K. Menzel, A. P. Zeng and W. D. Deckwer, Microbial production of 1,3-propanediol, Appl. Microbial Biotech., 52 (1999), 289-297. doi: 10.3934/dcdsb.2004.4.1065.

[5]

P. Billingsley, Convergence of Probability Measures, JohnWiley & Sons, New York, NY, USA, 1968. doi: 10.1007/978-1-4612-0873-0.

[6]

C. X. Gao, K. Z. Li, E. M. Feng and Z. L. Xiu, Nonlinear impulsive system of fed-batch culture in fermentative production and its properties, Chaos Solutions Fractals, 28 (2006), 271-277. doi: 10.3934/dcdsb.2004.4.1065.

[7]

C. X. Gao, Y. H. Lang, E. M. Feng and Z. L. Xiu, Nonlinear impulsive system of microbial production in fed-batch culture and its optimal control, J. Appl. Math. Comput., 19 (2005), 203-214. doi: 10.3934/dcdsb.2004.4.1065.

[8]

S. H. Hou and K. H. Wong, Optimal impulsive control problem with application to human immunodeficiency virus treatment, J. Optim. Theory Appl., 151 (2011), 385-401. doi: 10.3934/dcdsb.2004.4.1065.

[9]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. doi: 10.1007/978-1-4612-0873-0.

[10]

M. Mccoy, Chemical makers try biotech paths, Chem. Eng. News, 76 (1998), 13-19. doi: 10.3934/dcdsb.2004.4.1065.

[11]

K. Menzel, A. P. Zeng and W. D. Deckwer, High concentration and productivity of 1,3-propanediol from continuous fermentation of glycerol by Klebsiella pneumoniae, Enzyme Microb. Technol., 20 (1997), 82-86. doi: 10.3934/dcdsb.2004.4.1065.

[12]

C. Liu, Z. Gong and E. Feng, Optimal control for a nonlinear time-delay system in fed-batch fermentation, Pac. J. Optim., 9 (2013), 595-612. doi: 10.3934/dcdsb.2004.4.1065.

[13]

C. Liu and Z. Gong, Modelling and optimal control of a time-delayed switched system in fed-batch process, J. Franklin Inst., 35 (2014), 840-856. doi: 10.3934/dcdsb.2004.4.1065.

[14]

C. Liu, Sensitivity analysis and parameter identification for a nonlinear time-delay system in microbial fed-batch process, Appl. Math. Model., 38(2014), 1449-1463. doi: 10.3934/dcdsb.2004.4.1065.

[15]

Y. Liu, K. L. Teo, L. S. Jennings and S. Wang, On a class of optimal control problems with state jumps, J. Optim. Theory Appl., 98(1) (1998), 65-82. doi: 10.3934/dcdsb.2004.4.1065.

[16]

H. Y. Wang, E. M. Fenga and Z. L. Xiu, Optimality condition of the nonlinear impulsive system in fed-batch fermentation, Nonlinear Anal. TMA, 68, (2008), 12-23. doi: 10.3934/dcdsb.2004.4.1065.

[17]

L. Wang, Modelling and regularity of nonlinear impulsive switching dynamical system in fed-batch culture, Abstr. Appl. Anal., Volume 2012, Article ID 295627, 15 pages. doi: 10.3934/dcdsb.2004.4.1065.

[18]

Z. L. Xiu, A. P. Zeng and W. D. Deckwer, Multiplicity and stability analysis of microorganisms in continuous culture: effects of metabolic overflow and growth inhibition, Biotechnol. Bioeng., 57 (1998), 251-261. doi: 10.3934/dcdsb.2004.4.1065.

[19]

Z. L. Xiu, A. P. Zeng and L. J. An, Mathematical modeling of kinetics and research on multiplicity of glycerol bioconversion to 1,3-propanediol, J. Dalian Univ. of Technol., 44 (2000), 428-433. doi: 10.3934/dcdsb.2004.4.1065.

[20]

A. P. Zeng and H. Biebl, Bulk-chemicals from biotechnology: the case of microbial production of 1,3-propanediol and the new trends, Adv. Biochem. Eng. Biotechnol., 74 (2002), 239-259. doi: 10.3934/dcdsb.2004.4.1065.

[21]

A. P. Zeng, A. Ross, H. Biebl, C. Tag, B. Günzel and W. D. Deckwer, Multiple product inhibition and growth modeling of Clostridium butyricum and Klebsiella pneumoniae in glycerol fermentation, Biotechnol. Bioeng., 44 (1994), 902-911. doi: 10.3934/dcdsb.2004.4.1065.

show all references

References:
[1]

J. Angelova and A. Dishliev, Optimization problems for one-impulsive models from population dynamics, Nonlinear Anal., 39 (2000) 483-497. doi: 10.3934/dcdsb.2004.4.1065.

[2]

S. A. Attia, V. Azhmyakov and J. Raisch, On an optimization problem for a class of impulsive hybrid systems, Discrete Event. Dyn. Syst., 20 (2010), 215-231. doi: 10.3934/dcdsb.2004.4.1065.

[3]

V. Azhmyakov, V. G. Boltyanski and A. Poznyak, Optimal control of impulsive hybrid systems, Nonlinear Anal. Hyb. Syst., 2 (2008), 1089-1097. doi: 10.3934/dcdsb.2004.4.1065.

[4]

H. Biebl, K. Menzel, A. P. Zeng and W. D. Deckwer, Microbial production of 1,3-propanediol, Appl. Microbial Biotech., 52 (1999), 289-297. doi: 10.3934/dcdsb.2004.4.1065.

[5]

P. Billingsley, Convergence of Probability Measures, JohnWiley & Sons, New York, NY, USA, 1968. doi: 10.1007/978-1-4612-0873-0.

[6]

C. X. Gao, K. Z. Li, E. M. Feng and Z. L. Xiu, Nonlinear impulsive system of fed-batch culture in fermentative production and its properties, Chaos Solutions Fractals, 28 (2006), 271-277. doi: 10.3934/dcdsb.2004.4.1065.

[7]

C. X. Gao, Y. H. Lang, E. M. Feng and Z. L. Xiu, Nonlinear impulsive system of microbial production in fed-batch culture and its optimal control, J. Appl. Math. Comput., 19 (2005), 203-214. doi: 10.3934/dcdsb.2004.4.1065.

[8]

S. H. Hou and K. H. Wong, Optimal impulsive control problem with application to human immunodeficiency virus treatment, J. Optim. Theory Appl., 151 (2011), 385-401. doi: 10.3934/dcdsb.2004.4.1065.

[9]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. doi: 10.1007/978-1-4612-0873-0.

[10]

M. Mccoy, Chemical makers try biotech paths, Chem. Eng. News, 76 (1998), 13-19. doi: 10.3934/dcdsb.2004.4.1065.

[11]

K. Menzel, A. P. Zeng and W. D. Deckwer, High concentration and productivity of 1,3-propanediol from continuous fermentation of glycerol by Klebsiella pneumoniae, Enzyme Microb. Technol., 20 (1997), 82-86. doi: 10.3934/dcdsb.2004.4.1065.

[12]

C. Liu, Z. Gong and E. Feng, Optimal control for a nonlinear time-delay system in fed-batch fermentation, Pac. J. Optim., 9 (2013), 595-612. doi: 10.3934/dcdsb.2004.4.1065.

[13]

C. Liu and Z. Gong, Modelling and optimal control of a time-delayed switched system in fed-batch process, J. Franklin Inst., 35 (2014), 840-856. doi: 10.3934/dcdsb.2004.4.1065.

[14]

C. Liu, Sensitivity analysis and parameter identification for a nonlinear time-delay system in microbial fed-batch process, Appl. Math. Model., 38(2014), 1449-1463. doi: 10.3934/dcdsb.2004.4.1065.

[15]

Y. Liu, K. L. Teo, L. S. Jennings and S. Wang, On a class of optimal control problems with state jumps, J. Optim. Theory Appl., 98(1) (1998), 65-82. doi: 10.3934/dcdsb.2004.4.1065.

[16]

H. Y. Wang, E. M. Fenga and Z. L. Xiu, Optimality condition of the nonlinear impulsive system in fed-batch fermentation, Nonlinear Anal. TMA, 68, (2008), 12-23. doi: 10.3934/dcdsb.2004.4.1065.

[17]

L. Wang, Modelling and regularity of nonlinear impulsive switching dynamical system in fed-batch culture, Abstr. Appl. Anal., Volume 2012, Article ID 295627, 15 pages. doi: 10.3934/dcdsb.2004.4.1065.

[18]

Z. L. Xiu, A. P. Zeng and W. D. Deckwer, Multiplicity and stability analysis of microorganisms in continuous culture: effects of metabolic overflow and growth inhibition, Biotechnol. Bioeng., 57 (1998), 251-261. doi: 10.3934/dcdsb.2004.4.1065.

[19]

Z. L. Xiu, A. P. Zeng and L. J. An, Mathematical modeling of kinetics and research on multiplicity of glycerol bioconversion to 1,3-propanediol, J. Dalian Univ. of Technol., 44 (2000), 428-433. doi: 10.3934/dcdsb.2004.4.1065.

[20]

A. P. Zeng and H. Biebl, Bulk-chemicals from biotechnology: the case of microbial production of 1,3-propanediol and the new trends, Adv. Biochem. Eng. Biotechnol., 74 (2002), 239-259. doi: 10.3934/dcdsb.2004.4.1065.

[21]

A. P. Zeng, A. Ross, H. Biebl, C. Tag, B. Günzel and W. D. Deckwer, Multiple product inhibition and growth modeling of Clostridium butyricum and Klebsiella pneumoniae in glycerol fermentation, Biotechnol. Bioeng., 44 (1994), 902-911. doi: 10.3934/dcdsb.2004.4.1065.

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