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An almost periodic Dengue transmission model with age structure and time-delayed input of vector in a patchy environment
Optimal control of leachate recirculation for anaerobic processes in landfills
Department of Mathematical, Physical and Computer Sciences, University of Parma, Parco Area delle Scienze 53/A, 43124, Parma, Italy |
A mathematical model for the degradation of the organic fraction of solid waste in landfills, by means of an anaerobic bacterial population, is proposed. Additional phenomena, like hydrolysis of insoluble substrate and biomass decay, are taken into account. The evolution of the system is monitored by controlling the effects of leachate recirculation on the hydrolytic process. We investigate the optimal strategies to minimize substrate concentration and recirculation operation costs. Analytical and numerical results are presented and discussed for linear and quadratic cost functionals.
References:
[1] |
J. F. Andrews,
A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol. Bioeng., 10 (1968), 707-723.
doi: 10.1002/bit.260100602. |
[2] |
S. Anita, V. Capasso and V. Arnautu, An Introduction to Optimal Control Problems in Life Sciences and Economics: From Mathematical Models to Numerical Simulation with MATLAB, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser/Springer, New York, 2011.
doi: 10.1007/978-0-8176-8098-5. |
[3] |
O. Bara, S. M. Djouadi, J. D. Day and S. Lenhart,
Immune therapeutic strategies using optimal controls with L1 and L2 type objectives, Math. Biosci., 290 (2017), 9-21.
doi: 10.1016/j.mbs.2017.05.010. |
[4] |
T. Bayen, O. Cots and P. Gajardo,
Analysis of an optimal control problem related to anaerobic digestion process, J. Optimiz. Theory App., 178 (2018), 627-659.
doi: 10.1007/s10957-018-1292-7. |
[5] |
T. Bayen and P. Gajardo,
On the steady state optimization of the biogas production in a two-stage anaerobic digestion model, J. Math. Biol., 78 (2019), 1067-1087.
doi: 10.1007/s00285-018-1301-3. |
[6] |
T. Bayen, J. Harmand and M. Sebbah,
Time-optimal control of concentration changes in the chemostat with one single species, Appl. Math. Model., 50 (2017), 257-278.
doi: 10.1016/j.apm.2017.05.037. |
[7] |
S. Bozkurt, L. Moreno and I. Neretnieks,
Long-term processes in waste deposits, Sci. Total Environ., 250 (2000), 101-121.
doi: 10.1016/S0048-9697(00)00370-3. |
[8] |
S.-J. Feng, B.-Y. Cao and H.-J. Xie, Modeling of leachate recirculation using spraying-vertical well systems in bioreactor landfills, Int. J. Geomech., 17 (2017), 04017012.
doi: 10.1061/(ASCE)GM.1943-5622.0000887. |
[9] |
H. V. M. Hamelers, A Mathematical Model for Composting Kinetics, Ph.D thesis, Wageningen University, 2001. Google Scholar |
[10] |
J. Harmsen,
Identification of organic compounds in leachate from a waste tip, Water Res., 17 (1983), 699-705.
doi: 10.1016/0043-1354(83)90239-7. |
[11] |
R. T. Haug, The Practical Handbook of Compost Engineering, Lewis Publishers, Boca Raton, FL, 1993.
doi: 10.1201/9780203736234. |
[12] |
M. M. Haydar and M. V Khire,
Leachate recirculation using permeable blankets in engineered landfills, J. Geotech. GeoEnviron., 133 (2007), 360-371.
doi: 10.1061/(ASCE)1090-0241(2007)133:4(360). |
[13] |
A. Husain,
Mathematical models of the kinetics of anaerobic digestion - a selected review, Biomass Bioenergy, 14 (1998), 561-571.
doi: 10.1016/S0961-9534(97)10047-2. |
[14] |
U. Ledzewicz, T. Brown and H. Schättler,
Comparison of optimal controls for a model in cancer chemotherapy with $L_1$- and $L_2$-type objectives, Optim. Method. Softw., 19 (2004), 339-350.
doi: 10.1080/10556780410001683104. |
[15] |
P. J. Maris, D. W. Harrington and F. E. Mosey,
Treatment of landfill leachate; management options, Water Qual. Res. J. Can., 20 (1985), 25-42.
doi: 10.2166/wqrj.1985.026. |
[16] |
G. Martalò, C. Bianchi, B. Buonomo, M. Chiappini and V. Vespri,
Mathematical modeling of oxygen control in biocell composting plants, Math. Comput. Simulat., 177 (2020), 105-119.
doi: 10.1016/j.matcom.2020.04.011. |
[17] |
G. Martalò, C. Bianchi, B. Buonomo, M. Chiappini and V. Vespri, On the role of inhibition processes in modeling control strategies for composting plants, in Current Trends in Dynamical Systems in Biology and Natural Sciences. SEMA SIMAI Springer Series, Volume 21 (editors M. Aguiar, C. Braumann, B. Kooi, A. Pugliese, N. Stollenwerk and E. Venturino), Springer, Cham, (2020), 125–145. Google Scholar |
[18] |
J. Monod,
The growth of bacterial cultures, Annu. Rev. Microbiol., 3 (1949), 371-394.
doi: 10.1146/annurev.mi.03.100149.002103. |
[19] |
J. Nygren, Output Feedback Control: Some Methods and Applications, Doctoral dissertation, Uppsala Universitet, 2014. Google Scholar |
[20] |
J. Pacey, D. Augenstein, R. Morck, D. Reinhart and R. Yazdani, The bioreactor landfill - An innovation in solid waste management, MSW Management (1999). Available from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.460.2984&rep=rep1&type=pdf Google Scholar |
[21] |
L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4684-0392-3. |
[22] |
C. Polprasert, Organic Waste Recycling, John Wiley and Sons Ltd., 1989. Google Scholar |
[23] | L. S. Pontryagin, Mathematical Theory of Optimal Processes, CRC Press, 1987. Google Scholar |
[24] |
X. Qian, R. M. Koerner and D. H. Gray, Geotechnical Aspects of Landfill Design and Construction, Pearson College Div., 2001. Google Scholar |
[25] |
A. Rapaport, T. Bayen, M. Sebbah, A. Donoso-Bravo and A. Torrico,
Dynamical modeling and optimal control of landfills, Math. Models Methods Appl. Sci., 26 (2016), 901-929.
doi: 10.1142/S0218202516500214. |
[26] |
S. Revollar, P. Vega, R. Vilanova and M. Francisco, Optimal control of wastewater treatment plants using economic-oriented model predictive dynamic strategies, App. Sci., 7 (2017), 813.
doi: 10.3390/app7080813. |
[27] |
H. Schaettler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer Science & Business Media, Springer, New York, 2012.
doi: 10.1007/978-1-4614-3834-2. |
[28] |
D. T. Sponza and O. N. Agdag,
Impact of leachate recirculation and recirculation volume on stabilization of municipal solid wastes in simulated anaerobic bioreactors, Process Biochem., 39 (2004), 2157-2165.
doi: 10.1016/j.procbio.2003.11.012. |
[29] |
R. F. Stengel, R. Ghigliazza, N. Kulkarni and O. Laplace,
Optimal control of innate immune response, Optim. Control Appl. Meth., 23 (2002), 91-104.
doi: 10.1002/oca.704. |
[30] |
V. A. Vavilin, S. V. Rytov, L. Y. Lokshina, S. G. Pavlostathis and M. A. Barlaz,
Distributed model of solid waste anaerobic digestion: effects of leachate recirculation and pH adjustment, Biotechnol. Bioeng., 81 (2003), 66-73.
doi: 10.1002/bit.10450. |
show all references
References:
[1] |
J. F. Andrews,
A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol. Bioeng., 10 (1968), 707-723.
doi: 10.1002/bit.260100602. |
[2] |
S. Anita, V. Capasso and V. Arnautu, An Introduction to Optimal Control Problems in Life Sciences and Economics: From Mathematical Models to Numerical Simulation with MATLAB, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser/Springer, New York, 2011.
doi: 10.1007/978-0-8176-8098-5. |
[3] |
O. Bara, S. M. Djouadi, J. D. Day and S. Lenhart,
Immune therapeutic strategies using optimal controls with L1 and L2 type objectives, Math. Biosci., 290 (2017), 9-21.
doi: 10.1016/j.mbs.2017.05.010. |
[4] |
T. Bayen, O. Cots and P. Gajardo,
Analysis of an optimal control problem related to anaerobic digestion process, J. Optimiz. Theory App., 178 (2018), 627-659.
doi: 10.1007/s10957-018-1292-7. |
[5] |
T. Bayen and P. Gajardo,
On the steady state optimization of the biogas production in a two-stage anaerobic digestion model, J. Math. Biol., 78 (2019), 1067-1087.
doi: 10.1007/s00285-018-1301-3. |
[6] |
T. Bayen, J. Harmand and M. Sebbah,
Time-optimal control of concentration changes in the chemostat with one single species, Appl. Math. Model., 50 (2017), 257-278.
doi: 10.1016/j.apm.2017.05.037. |
[7] |
S. Bozkurt, L. Moreno and I. Neretnieks,
Long-term processes in waste deposits, Sci. Total Environ., 250 (2000), 101-121.
doi: 10.1016/S0048-9697(00)00370-3. |
[8] |
S.-J. Feng, B.-Y. Cao and H.-J. Xie, Modeling of leachate recirculation using spraying-vertical well systems in bioreactor landfills, Int. J. Geomech., 17 (2017), 04017012.
doi: 10.1061/(ASCE)GM.1943-5622.0000887. |
[9] |
H. V. M. Hamelers, A Mathematical Model for Composting Kinetics, Ph.D thesis, Wageningen University, 2001. Google Scholar |
[10] |
J. Harmsen,
Identification of organic compounds in leachate from a waste tip, Water Res., 17 (1983), 699-705.
doi: 10.1016/0043-1354(83)90239-7. |
[11] |
R. T. Haug, The Practical Handbook of Compost Engineering, Lewis Publishers, Boca Raton, FL, 1993.
doi: 10.1201/9780203736234. |
[12] |
M. M. Haydar and M. V Khire,
Leachate recirculation using permeable blankets in engineered landfills, J. Geotech. GeoEnviron., 133 (2007), 360-371.
doi: 10.1061/(ASCE)1090-0241(2007)133:4(360). |
[13] |
A. Husain,
Mathematical models of the kinetics of anaerobic digestion - a selected review, Biomass Bioenergy, 14 (1998), 561-571.
doi: 10.1016/S0961-9534(97)10047-2. |
[14] |
U. Ledzewicz, T. Brown and H. Schättler,
Comparison of optimal controls for a model in cancer chemotherapy with $L_1$- and $L_2$-type objectives, Optim. Method. Softw., 19 (2004), 339-350.
doi: 10.1080/10556780410001683104. |
[15] |
P. J. Maris, D. W. Harrington and F. E. Mosey,
Treatment of landfill leachate; management options, Water Qual. Res. J. Can., 20 (1985), 25-42.
doi: 10.2166/wqrj.1985.026. |
[16] |
G. Martalò, C. Bianchi, B. Buonomo, M. Chiappini and V. Vespri,
Mathematical modeling of oxygen control in biocell composting plants, Math. Comput. Simulat., 177 (2020), 105-119.
doi: 10.1016/j.matcom.2020.04.011. |
[17] |
G. Martalò, C. Bianchi, B. Buonomo, M. Chiappini and V. Vespri, On the role of inhibition processes in modeling control strategies for composting plants, in Current Trends in Dynamical Systems in Biology and Natural Sciences. SEMA SIMAI Springer Series, Volume 21 (editors M. Aguiar, C. Braumann, B. Kooi, A. Pugliese, N. Stollenwerk and E. Venturino), Springer, Cham, (2020), 125–145. Google Scholar |
[18] |
J. Monod,
The growth of bacterial cultures, Annu. Rev. Microbiol., 3 (1949), 371-394.
doi: 10.1146/annurev.mi.03.100149.002103. |
[19] |
J. Nygren, Output Feedback Control: Some Methods and Applications, Doctoral dissertation, Uppsala Universitet, 2014. Google Scholar |
[20] |
J. Pacey, D. Augenstein, R. Morck, D. Reinhart and R. Yazdani, The bioreactor landfill - An innovation in solid waste management, MSW Management (1999). Available from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.460.2984&rep=rep1&type=pdf Google Scholar |
[21] |
L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4684-0392-3. |
[22] |
C. Polprasert, Organic Waste Recycling, John Wiley and Sons Ltd., 1989. Google Scholar |
[23] | L. S. Pontryagin, Mathematical Theory of Optimal Processes, CRC Press, 1987. Google Scholar |
[24] |
X. Qian, R. M. Koerner and D. H. Gray, Geotechnical Aspects of Landfill Design and Construction, Pearson College Div., 2001. Google Scholar |
[25] |
A. Rapaport, T. Bayen, M. Sebbah, A. Donoso-Bravo and A. Torrico,
Dynamical modeling and optimal control of landfills, Math. Models Methods Appl. Sci., 26 (2016), 901-929.
doi: 10.1142/S0218202516500214. |
[26] |
S. Revollar, P. Vega, R. Vilanova and M. Francisco, Optimal control of wastewater treatment plants using economic-oriented model predictive dynamic strategies, App. Sci., 7 (2017), 813.
doi: 10.3390/app7080813. |
[27] |
H. Schaettler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer Science & Business Media, Springer, New York, 2012.
doi: 10.1007/978-1-4614-3834-2. |
[28] |
D. T. Sponza and O. N. Agdag,
Impact of leachate recirculation and recirculation volume on stabilization of municipal solid wastes in simulated anaerobic bioreactors, Process Biochem., 39 (2004), 2157-2165.
doi: 10.1016/j.procbio.2003.11.012. |
[29] |
R. F. Stengel, R. Ghigliazza, N. Kulkarni and O. Laplace,
Optimal control of innate immune response, Optim. Control Appl. Meth., 23 (2002), 91-104.
doi: 10.1002/oca.704. |
[30] |
V. A. Vavilin, S. V. Rytov, L. Y. Lokshina, S. G. Pavlostathis and M. A. Barlaz,
Distributed model of solid waste anaerobic digestion: effects of leachate recirculation and pH adjustment, Biotechnol. Bioeng., 81 (2003), 66-73.
doi: 10.1002/bit.10450. |














0.1 | 5.03 | 0.4031 | 0.1454 |
0.3 | 5.10 | 0.4043 | 0.1428 |
0.5 | 5.25 | 0.4063 | 0.1378 |
0.7 | 5.58 | 0.4084 | 0.1262 |
0.1 | 5.03 | 0.4031 | 0.1454 |
0.3 | 5.10 | 0.4043 | 0.1428 |
0.5 | 5.25 | 0.4063 | 0.1378 |
0.7 | 5.58 | 0.4084 | 0.1262 |
0.1 | 5.25 | 0.4063 | 0.1378 |
0.2 | 5.16 | 0.4046 | 0.1406 |
0.3 | 5.10 | 0.4021 | 0.1422 |
0.4 | 5.09 | 0.3933 | 0.1414 |
0.1 | 5.25 | 0.4063 | 0.1378 |
0.2 | 5.16 | 0.4046 | 0.1406 |
0.3 | 5.10 | 0.4021 | 0.1422 |
0.4 | 5.09 | 0.3933 | 0.1414 |
0.1 | 5.72 | 0.4008 | 0.1210 |
0.3 | 5.78 | 0.4012 | 0.1192 |
0.5 | 5.88 | 0.4018 | 0.1157 |
0.7 | 6.16 | 0.4010 | 0.1075 |
0.1 | 5.72 | 0.4008 | 0.1210 |
0.3 | 5.78 | 0.4012 | 0.1192 |
0.5 | 5.88 | 0.4018 | 0.1157 |
0.7 | 6.16 | 0.4010 | 0.1075 |
0.1 | 5.88 | 0.4018 | 0.1157 |
0.2 | 5.84 | 0.4009 | 0.1171 |
0.3 | 5.86 | 0.3990 | 0.1169 |
0.4 | 6.04 | 0.3902 | 0.1135 |
0.1 | 5.88 | 0.4018 | 0.1157 |
0.2 | 5.84 | 0.4009 | 0.1171 |
0.3 | 5.86 | 0.3990 | 0.1169 |
0.4 | 6.04 | 0.3902 | 0.1135 |
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