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July  2013, 9(3): 505-524. doi: 10.3934/jimo.2013.9.505

Control parametrization and finite element method for controlling multi-species reactive transport in a circular pool

Heung Wing Joseph Lee 1, , Chi Kin Chan 1, , Karho Yau 1, , Kar Hung Wong 2, and  Colin Myburgh 2,

1. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China, China
2. 

School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa, South Africa

Received  October 2011 Revised  October 2012 Published  April 2013

In this paper, we consider an optimal control problem for a cleaning program involving effluent discharge of several species in a circular pool. A computational scheme combining control parametrization and finite element method is used to develop a cleaning program to meet the environmental health requirements. A numerical example is solved to illustrate the efficiency of our method.
Keywords: Galerkin Scheme, Species concentration, control parametrization, circular pool, non-negativity requirement of state variables., optimal control.
Mathematics Subject Classification: Primary: 49M15, 65M60; Secondary: 35Q9.
Citation: Heung Wing Joseph Lee, Chi Kin Chan, Karho Yau, Kar Hung Wong, Colin Myburgh. Control parametrization and finite element method for controlling multi-species reactive transport in a circular pool. Journal of Industrial & Management Optimization, 2013, 9 (3) : 505-524. doi: 10.3934/jimo.2013.9.505
References:
[1]

R. C. Borden, C. A. Gomez and M. T. Becker, Geochemical indicators of intrinsic bioremediation, Ground Water, 33 (1995), 180-189. doi: 10.1111/j.1745-6584.1995.tb00272.x.  Google Scholar

[2]

T. P. Clement, Generalized solution to multispecies transport equations coupled with a first-order reaction network, Water Resources Research, 37 (2001), 157-163. doi: 10.1029/2000WR900239.  Google Scholar

[3]

T. P. Clement, C. D. Johnson, Y. Sun, G. M. Klecka and C. Bartlett, Natural attenuation of chlorinated ethene compounds: model development and field-scale application at the Dover site, Journal of Contaminant Hydrology, 42 (2000), 113-140. doi: 10.1016/S0169-7722(99)00098-4.  Google Scholar

[4]

T. P. Clement, Y. Sun, B. S. Hooker and J. N. Petersen, Modeling multispecies reactive transport in ground water, Ground Water Monitoring & Remediation, 18 (1998), 79-92. doi: 10.1111/j.1745-6592.1998.tb00618.x.  Google Scholar

[5]

M. Gerdts and M. Kunkel, A nonsmooth Newton's method for discretized optimal control problems with state and control constraints, Journal of Industrial and Management Optimization, 4 (2008), 247-270. doi: 10.3934/jimo.2008.4.247.  Google Scholar

[6]

L. S. Jennings, K. L. Teo, M. E. Fisher and C. J. Goh, MISER3 version 3, Optimal control software : Theory and user manual, Centre for Applied Dynamics and Optimization, The University of Western Australia, 2004. http://school.maths.uwa.edu.au/les/miser/manual.html. Google Scholar

[7]

K. Kaji and K. H. Wong, Nonlinearly constrained time-delayed optimal control problems, Journal of Optimization Theory and Applications, 82 (1994), 295-313. doi: 10.1007/BF02191855.  Google Scholar

[8]

H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings, Control parametrization enhancing technique for time optimal control problems, Dynamic Systems and Applications, 6 (1997), 243-262.  Google Scholar

[9]

M. S. Lee, K. K. Lee, Y. Hyun, T. P. Clement and D. Hamilton, Nitrogen transformation and transport modeling in groundwater aquifers, Ecological Modelling, 192 (2006), 143-159. doi: 10.1016/j.ecolmodel.2005.07.013.  Google Scholar

[10]

B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, Journal of Optimization Theory and Applications, 151 (2011), 260-291. doi: 10.1007/s10957-011-9904-5.  Google Scholar

[11]

Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for a class of free terminal time optimal control problems, Pacific Journal of Optimization, 7 (2011), 63-81.  Google Scholar

[12]

R. C. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44 (2008), 2923-2929. doi: 10.1016/j.automatica.2008.04.011.  Google Scholar

[13]

M. Lunn, R. J. Lunn and R. Mackayb, Determining analytic solutions of multiple species contaminant transport, with sorption and decay, Journal of Hydrology, 180 (1996), 195-210. doi: 10.1016/0022-1694(95)02891-9.  Google Scholar

[14]

H. Maurer, C. Büshens, J. H. R. Kim and C. Y. Kaya, Optimization methods for the verification of second order sufficient conditions for bang-bang controls, Optimal Control Applications and Methods, 26 (2005), 129-156. doi: 10.1002/oca.756.  Google Scholar

[15]

D. E. Rice, R. D. Grose, J. C. Michaelsen, B. P. Dooher, D. H. Macqueen, S. J. Cullen, W. E. Kastenberg, L. G. Everett and M. S. Marino, "California Leaking Underground Fuel Tank (LUFT) Historical Case Analyses," California State Water Resources Publication, UCRL-AR-122206, 1995. Google Scholar

[16]

L. Semprini, P. K. Kitanidis, D. H. Kampbell and J. T. Wilson, Anaerobic transformation of chlorinated aliphatic hydrocarbons in a sand aquifer based on spatial chemical distributions, Water Resources Research, 31 (1995), 1051-1062. doi: 10.1029/94WR02380.  Google Scholar

[17]

H. Tao and X. Liu, An improved control parameterization method for chemical dynamic optimization problems, World Congress on Intelligent Control and Automation, WCICA, (2006), 1650-1653. Google Scholar

[18]

K. L. Teo, C. J. Goh and K. H. Wong, "A Unified Computational Approach to Optimal Control Problems," Pitman Monographs and Surveys in Pure and Applied Mathematics 55, Longman Scientific & Technical, 1991.  Google Scholar

[19]

K. L. Teo, L. S. Jennings, H. W. J. Lee and V. Rehbock, The control parameterization enhancing transform for constrained optimal control problems, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 40 (1999), 314-335. doi: 10.1017/S0334270000010936.  Google Scholar

[20]

K. L. Teo, H. W. J. Lee and V. Rehbock, Control parametrization enhancing technique for time optimal control and optimal three-valued control problems, Dynamics of Continuous, Discrete and Impulsive Systems, 4 (1998), 617-631.  Google Scholar

[21]

K. L. Teo, K. H. Wong and D. J. Clements, Optimal control computation for linear time-lag systems with linear terminal constraints, Journal of Optimization Theory and Applications, 44 (1984), 509-526. doi: 10.1007/BF00935465.  Google Scholar

[22]

L. Y. Wang, W. H. Gui, K. L. Teo, R. C. Loxton and C. H. Yang, Time-delay optimal control problems with multiple characteristic time points: Computation and industrial applications, Journal of Industrial and Management Optimization, 5 (2009), 705-718. doi: 10.3934/jimo.2009.5.705.  Google Scholar

[23]

K. H. Wong, D. J. Clements and K. L. Teo, Optimal control computation for nonlinear time-lag systems, Journal of Optimization Theory and Applications, 47 (1985), 91-107. doi: 10.1007/BF00941318.  Google Scholar

[24]

K. H. Wong, L. S. Jennings and F. Benyah, Control parametrization method for free planning time optimal control problems with time-delayed arguments, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 47 (2001), 5679-5690. doi: 10.1016/S0362-546X(01)00669-1.  Google Scholar

[25]

K. H. Wong, H. W. J. Lee and C. K. Chan, Control parametrization and finite element method for controlling multi-species reactive transport in a rectangular diffuser unit, Journal of Optimization Theory and Applications, 150 (2011), 118-141. doi: 10.1007/s10957-011-9826-2.  Google Scholar

show all references

References:
[1]

R. C. Borden, C. A. Gomez and M. T. Becker, Geochemical indicators of intrinsic bioremediation, Ground Water, 33 (1995), 180-189. doi: 10.1111/j.1745-6584.1995.tb00272.x.  Google Scholar

[2]

T. P. Clement, Generalized solution to multispecies transport equations coupled with a first-order reaction network, Water Resources Research, 37 (2001), 157-163. doi: 10.1029/2000WR900239.  Google Scholar

[3]

T. P. Clement, C. D. Johnson, Y. Sun, G. M. Klecka and C. Bartlett, Natural attenuation of chlorinated ethene compounds: model development and field-scale application at the Dover site, Journal of Contaminant Hydrology, 42 (2000), 113-140. doi: 10.1016/S0169-7722(99)00098-4.  Google Scholar

[4]

T. P. Clement, Y. Sun, B. S. Hooker and J. N. Petersen, Modeling multispecies reactive transport in ground water, Ground Water Monitoring & Remediation, 18 (1998), 79-92. doi: 10.1111/j.1745-6592.1998.tb00618.x.  Google Scholar

[5]

M. Gerdts and M. Kunkel, A nonsmooth Newton's method for discretized optimal control problems with state and control constraints, Journal of Industrial and Management Optimization, 4 (2008), 247-270. doi: 10.3934/jimo.2008.4.247.  Google Scholar

[6]

L. S. Jennings, K. L. Teo, M. E. Fisher and C. J. Goh, MISER3 version 3, Optimal control software : Theory and user manual, Centre for Applied Dynamics and Optimization, The University of Western Australia, 2004. http://school.maths.uwa.edu.au/les/miser/manual.html. Google Scholar

[7]

K. Kaji and K. H. Wong, Nonlinearly constrained time-delayed optimal control problems, Journal of Optimization Theory and Applications, 82 (1994), 295-313. doi: 10.1007/BF02191855.  Google Scholar

[8]

H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings, Control parametrization enhancing technique for time optimal control problems, Dynamic Systems and Applications, 6 (1997), 243-262.  Google Scholar

[9]

M. S. Lee, K. K. Lee, Y. Hyun, T. P. Clement and D. Hamilton, Nitrogen transformation and transport modeling in groundwater aquifers, Ecological Modelling, 192 (2006), 143-159. doi: 10.1016/j.ecolmodel.2005.07.013.  Google Scholar

[10]

B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, Journal of Optimization Theory and Applications, 151 (2011), 260-291. doi: 10.1007/s10957-011-9904-5.  Google Scholar

[11]

Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for a class of free terminal time optimal control problems, Pacific Journal of Optimization, 7 (2011), 63-81.  Google Scholar

[12]

R. C. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44 (2008), 2923-2929. doi: 10.1016/j.automatica.2008.04.011.  Google Scholar

[13]

M. Lunn, R. J. Lunn and R. Mackayb, Determining analytic solutions of multiple species contaminant transport, with sorption and decay, Journal of Hydrology, 180 (1996), 195-210. doi: 10.1016/0022-1694(95)02891-9.  Google Scholar

[14]

H. Maurer, C. Büshens, J. H. R. Kim and C. Y. Kaya, Optimization methods for the verification of second order sufficient conditions for bang-bang controls, Optimal Control Applications and Methods, 26 (2005), 129-156. doi: 10.1002/oca.756.  Google Scholar

[15]

D. E. Rice, R. D. Grose, J. C. Michaelsen, B. P. Dooher, D. H. Macqueen, S. J. Cullen, W. E. Kastenberg, L. G. Everett and M. S. Marino, "California Leaking Underground Fuel Tank (LUFT) Historical Case Analyses," California State Water Resources Publication, UCRL-AR-122206, 1995. Google Scholar

[16]

L. Semprini, P. K. Kitanidis, D. H. Kampbell and J. T. Wilson, Anaerobic transformation of chlorinated aliphatic hydrocarbons in a sand aquifer based on spatial chemical distributions, Water Resources Research, 31 (1995), 1051-1062. doi: 10.1029/94WR02380.  Google Scholar

[17]

H. Tao and X. Liu, An improved control parameterization method for chemical dynamic optimization problems, World Congress on Intelligent Control and Automation, WCICA, (2006), 1650-1653. Google Scholar

[18]

K. L. Teo, C. J. Goh and K. H. Wong, "A Unified Computational Approach to Optimal Control Problems," Pitman Monographs and Surveys in Pure and Applied Mathematics 55, Longman Scientific & Technical, 1991.  Google Scholar

[19]

K. L. Teo, L. S. Jennings, H. W. J. Lee and V. Rehbock, The control parameterization enhancing transform for constrained optimal control problems, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 40 (1999), 314-335. doi: 10.1017/S0334270000010936.  Google Scholar

[20]

K. L. Teo, H. W. J. Lee and V. Rehbock, Control parametrization enhancing technique for time optimal control and optimal three-valued control problems, Dynamics of Continuous, Discrete and Impulsive Systems, 4 (1998), 617-631.  Google Scholar

[21]

K. L. Teo, K. H. Wong and D. J. Clements, Optimal control computation for linear time-lag systems with linear terminal constraints, Journal of Optimization Theory and Applications, 44 (1984), 509-526. doi: 10.1007/BF00935465.  Google Scholar

[22]

L. Y. Wang, W. H. Gui, K. L. Teo, R. C. Loxton and C. H. Yang, Time-delay optimal control problems with multiple characteristic time points: Computation and industrial applications, Journal of Industrial and Management Optimization, 5 (2009), 705-718. doi: 10.3934/jimo.2009.5.705.  Google Scholar

[23]

K. H. Wong, D. J. Clements and K. L. Teo, Optimal control computation for nonlinear time-lag systems, Journal of Optimization Theory and Applications, 47 (1985), 91-107. doi: 10.1007/BF00941318.  Google Scholar

[24]

K. H. Wong, L. S. Jennings and F. Benyah, Control parametrization method for free planning time optimal control problems with time-delayed arguments, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 47 (2001), 5679-5690. doi: 10.1016/S0362-546X(01)00669-1.  Google Scholar

[25]

K. H. Wong, H. W. J. Lee and C. K. Chan, Control parametrization and finite element method for controlling multi-species reactive transport in a rectangular diffuser unit, Journal of Optimization Theory and Applications, 150 (2011), 118-141. doi: 10.1007/s10957-011-9826-2.  Google Scholar

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