
-
Previous Article
Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations
- DCDS-B Home
- This Issue
-
Next Article
Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials
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. |
[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. |
[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. |
[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 |
[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. |
[23] |
L. S. Pontryagin, Mathematical Theory of Optimal Processes, CRC Press, 1987.
![]() |
[24] |
X. Qian, R. M. Koerner and D. H. Gray, Geotechnical Aspects of Landfill Design and Construction, Pearson College Div., 2001. |
[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. |
[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. |
[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. |
[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 |
[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. |
[23] |
L. S. Pontryagin, Mathematical Theory of Optimal Processes, CRC Press, 1987.
![]() |
[24] |
X. Qian, R. M. Koerner and D. H. Gray, Geotechnical Aspects of Landfill Design and Construction, Pearson College Div., 2001. |
[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 |
[1] |
Andrew J. Whittle, Suzanne Lenhart, Louis J. Gross. Optimal control for management of an invasive plant species. Mathematical Biosciences & Engineering, 2007, 4 (1) : 101-112. doi: 10.3934/mbe.2007.4.101 |
[2] |
Yujing Wang, Changjun Yu, Kok Lay Teo. A new computational strategy for optimal control problem with a cost on changing control. Numerical Algebra, Control and Optimization, 2016, 6 (3) : 339-364. doi: 10.3934/naco.2016016 |
[3] |
C.E.M. Pearce, J. Piantadosi, P.G. Howlett. On an optimal control policy for stormwater management in two connected dams. Journal of Industrial and Management Optimization, 2007, 3 (2) : 313-320. doi: 10.3934/jimo.2007.3.313 |
[4] |
Mourad Azi, Mohand Ouamer Bibi. Optimal control of a dynamical system with intermediate phase constraints and applications in cash management. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 279-291. doi: 10.3934/naco.2021005 |
[5] |
Shuren Liu, Qiying Hu, Yifan Xu. Optimal inventory control with fixed ordering cost for selling by internet auctions. Journal of Industrial and Management Optimization, 2012, 8 (1) : 19-40. doi: 10.3934/jimo.2012.8.19 |
[6] |
Jésus Ildefonso Díaz, Tommaso Mingazzini, Ángel Manuel Ramos. On the optimal control for a semilinear equation with cost depending on the free boundary. Networks and Heterogeneous Media, 2012, 7 (4) : 605-615. doi: 10.3934/nhm.2012.7.605 |
[7] |
Kerem Uǧurlu. Continuity of cost functional and optimal feedback controls for the stochastic Navier Stokes equation in 2D. Communications on Pure and Applied Analysis, 2017, 16 (1) : 189-208. doi: 10.3934/cpaa.2017009 |
[8] |
Zhaohua Gong, Chongyang Liu, Yujing Wang. Optimal control of switched systems with multiple time-delays and a cost on changing control. Journal of Industrial and Management Optimization, 2018, 14 (1) : 183-198. doi: 10.3934/jimo.2017042 |
[9] |
Folashade B. Agusto. Optimal control and cost-effectiveness analysis of a three age-structured transmission dynamics of chikungunya virus. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 687-715. doi: 10.3934/dcdsb.2017034 |
[10] |
Jian-Xin Guo, Xing-Long Qu. Robust control in green production management. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1115-1132. doi: 10.3934/jimo.2021011 |
[11] |
Giuseppe Buttazzo, Serena Guarino Lo Bianco, Fabrizio Oliviero. Optimal location problems with routing cost. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1301-1317. doi: 10.3934/dcds.2014.34.1301 |
[12] |
Wei Xu, Liying Yu, Gui-Hua Lin, Zhi Guo Feng. Optimal switching signal design with a cost on switching action. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2531-2549. doi: 10.3934/jimo.2019068 |
[13] |
Lars Grüne, Marleen Stieler. Multiobjective model predictive control for stabilizing cost criteria. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3905-3928. doi: 10.3934/dcdsb.2018336 |
[14] |
Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2021, 13 (1) : 1-23. doi: 10.3934/jgm.2020032 |
[15] |
Julius Fergy T. Rabago, Hideyuki Azegami. A new energy-gap cost functional approach for the exterior Bernoulli free boundary problem. Evolution Equations and Control Theory, 2019, 8 (4) : 785-824. doi: 10.3934/eect.2019038 |
[16] |
Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial and Management Optimization, 2022, 18 (1) : 375-396. doi: 10.3934/jimo.2020158 |
[17] |
Zhengyan Wang, Guanghua Xu, Peibiao Zhao, Zudi Lu. The optimal cash holding models for stochastic cash management of continuous time. Journal of Industrial and Management Optimization, 2018, 14 (1) : 1-17. doi: 10.3934/jimo.2017034 |
[18] |
Ximin Huang, Na Song, Wai-Ki Ching, Tak-Kuen Siu, Ka-Fai Cedric Yiu. A real option approach to optimal inventory management of retail products. Journal of Industrial and Management Optimization, 2012, 8 (2) : 379-389. doi: 10.3934/jimo.2012.8.379 |
[19] |
Qian Zhao, Zhuo Jin, Jiaqin Wei. Optimal investment and dividend payment strategies with debt management and reinsurance. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1323-1348. doi: 10.3934/jimo.2018009 |
[20] |
Yu Yuan, Hui Mi. Robust optimal asset-liability management with penalization on ambiguity. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021121 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]