# American Institute of Mathematical Sciences

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

 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.
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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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