# American Institute of Mathematical Sciences

September  2017, 22(7): 2939-2969. doi: 10.3934/dcdsb.2017158

## Minimization of carbon abatement cost: Modeling, analysis and simulation

 1 School of Mathematical Sciences, Tongji University, 1239 Siping Rd, Yangpu Dist, Shanghai 200092, China 2 Department of Applied Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA

* Corresponding author

Received  September 2014 Revised  April 2017 Published  May 2017

Fund Project: This work is supported by National Natural Science Foundation of China (No. 11271287) and China Scholarship Council. Yang and Liang would like to thank the hospitalities of University of Notre Dame for their visiting where this work is partially carried out.

In this paper, we consider a problem of minimizing the carbon abatement cost of a country. Two models are built within the stochastic optimal control framework based on two types of abatement policies. The corresponding HJB equations are deduced, and the existence and uniqueness of their classical solutions are established by PDE methods. Using parameters in the models obtained from real data, we carried out numerical simulations via semi-implicit method. Then we discussed the properties of the optimal policies and minimal costs. Our results suggest that a country needs to keep a relatively low economy and population growth rate and keep a stable economy in order to reduce the total carbon abatement cost. In the long run, it's better for a country to seek for more efficient carbon abatement techniques and an environmentally friendly way of economic development.

Citation: Xiaoli Yang, Jin Liang, Bei Hu. Minimization of carbon abatement cost: Modeling, analysis and simulation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2939-2969. doi: 10.3934/dcdsb.2017158
##### References:
 [1] F. Ackerman and R. Bueno, Use of McKinsey abatement cost curves for climate economics modeling, 2011. Available from: http://www.sei-international.org/mediamanager/documents/Publications/Climate/sei-workingpaperus-1102.pdf. [2] F. Ackerman, E. A. Stanton and R. Bueno, CRED: A new model for climate and development, Ecol. Econ., 85 (2013), 166-176. [3] A. A. A. Almihoub, H. M. Mula and M. M. Rahman, Marginal abatement cost curves (MACCs): Important approaches to obtain (firm and sector) greenhouse gases (GHGs) reduction, Int. J. Econ. Financ., 5 (2013), 35-54. [4] R. Belauar, A. Fahim and N. Touzi, Optimal production policy under the carbon emission market, 2011. Available from: http://www.math.fsu.edu/~fahim/carbon.pdf. [5] C. Böhringer and T. F. Rutherford, Combining bottom-up and top-down, Energ. Econ., 30 (2008), 574-596. [6] R. Carmona, M. Fehr and J. Hinz, Optimal stochastic control and carbon price formation, SIAM J. Control. Optim., 48 (2009), 2168-2190.  doi: 10.1137/080736910. [7] R. Carmona, M. Fehr, J. Hinz and A. Porchet, Market design for emission trading schemes, SIAM Rev., 52 (2010), 403-452.  doi: 10.1137/080722813. [8] U. Cetin and M. Verschuere, Pricing and hedging in carbon emissions markets, Int. J. Theor. Appl. Financ., 12 (2009), 949-967.  doi: 10.1142/S0219024909005531. [9] M. Chamon and P. Mauro, Pricing growth-indexed bonds, J. Bank. Financ., 30 (2006), 3349-3366. [10] Y. Chen, Second-order Parabolic Partial Differential Equation Beijing University Press, Beijing, 2003. [11] E. Commission, EU action against climate change: The EU emissions trading scheme, 2009. Available from: http://www.ab.gov.tr/files/ardb/evt/1_avrupa_birligi/1_6_raporlar/1_3_diger/environment/eu_emmissions_trading_scheme.pdf. [12] E. Commission, The EU emissions trading system (EU ETS), 2013. Available from: https://ec.europa.eu/clima/sites/clima/files/factsheet_ets_en.pdf. [13] P. Criqui, S. Mima and L. Viguier, Marginal abatement costs of CO$_2$ emission reductions, geographical flexibility and concrete ceilings: An assessment using the POLES model, Energ. Policy, 27 (1999), 585-601. [14] T. Dietz and E. A. Rosa, Effects of population and affluence on CO$_2$ emissions, P. Natl. Acad. Sci. USA, 94 (1997), 175-179. [15] A. D. Ellerman and B. K. Buchner, The European Union Emissions Trading Scheme: Origins, allocation, and early results, REEP, 1 (2007), 66-87. [16] A. D. Ellerman and A. Decaux, Analysis of post-Kyoto CO2 emissions trading using marginal abatement curves, 1998. Available from: http://dspace.mit.edu/handle/1721.1/3608. [17] P. Enkvist, T. Nauclér and J. Rosander, A cost curve for greenhouse gas reduction, McKinsey Quarterly, 1 (2007), 35-45. [18] H. Fallman, Revision of the EU emissions trading system, Available from: http://ec.europa.eu/enterprise/sectors/chemicals/files/wg_7_8fer08/15fallmann_ets_en.pdf. [19] M. Fehr and J. Hinz, A quantitative approach to carbon price risk modeling, Available from: http://www.bbk.ac.uk/cfc/pdfs/conference\%20papers/Thurs/FehrHinz.pdf. [20] B. S. Fisher, N. Nakicenovic, K. Alfsen, J. C. Morlot, F. De La Chesnaye, J. Hourcade, K. Jiang, M. Kainuma, E. La Rovere, A. Matysek, A. Rana, K. Riahi, R. Richels, S. Rose, D. V. Vuuren and R. Warren, Issues Related to Mitigation in the Long-term Context In Climate Change 2007: Mitigation. Contribution of Working Group Ⅲ to the Fourth Assessment Report of the Inter-governmental Panel on Climate Change [B. Metz, O. R. Davidson, P. R. Bosch, R. Dave and L. A. Meyer] Cambridge University Press, Cambridge, 2007. [21] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions Springer-Verlag, New York, 1993. [22] P. A. Forsyth and G. Labahn, Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance, 2007. Available from: https://cs.uwaterloo.ca/~paforsyt/hjb.pdf. [23] C. Hepburn, Carbon trading: A review of the Kyoto mechanisms, Annu. Rev. Environ. Resour., 32 (2007), 375-393. [24] J. Hourcade, M. Jaccard, C. Bataille and F. Ghersi, Hybrid modeling: New answers to old challenges, Energy J., 2 (2006), 1-11. [25] B. Hu, Blow-up Theories for Semilinear Parabolic Equations Lecture Notes in Mathematics, 2018, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18460-4. [26] International Emissions Trading Association (IETA), The world’s carbon markets: A case study guide to emissions trading, 2013. Available from: http://www.edf.org/sites/default/files/EDF_IETA_Mexico_Case_Study_May_2013.pdf. [27] M. Jakob, Marginal costs and co-benefits of energy efficiency investments: the case of the Swiss residential sector, Energ. Policy, 34 (2006), 172-187. [28] L. Jiang, Mathematical Modeling and Methods for Option Pricing Translated from the 2003 Chinese original by Canguo Li, World Scientific Publishing Co. , Inc. , River Edge, NJ, 2005. doi: 10.1142/5855. [29] F. Kesicki, Marginal abatement cost curves for policy making--expert-based vs. model-based curves, P. IAEE Int. Conf., (2010), 1-19. [30] F. Kesicki, Marginal abatement cost curves: Combining energy system modeling and decomposition analysis, Environ. Model. Assess., 18 (2013), 27-37. [31] F. Kasicki and N. Strachan, Marginal abatement cost (MAC) curves: Confronting theory and practice, Environ. Sci. Policy, 14 (2011), 1195-1204.  doi: 10.1016/j.envsci.2011.08.004. [32] G. Klepper and S. Peterson, Marginal abatement cost curves in general equilibrium: The influence of world energy prices, Resour. Energy Econ., 28 (2006), 1-23. [33] S. Kruse, M. Meitner and M. Schröder, On the pricing of GDP-linked financial products, Appl. Financ. Econ., 15 (2005), 1125-1133. [34] G. M. Lieberman, Second Order Parabolic Differential Equations World Scientific Publishing Co. Pte. Ltd. , Singapore, 1996. doi: 10.1142/3302. [35] J. J. McCarthy, O. F. Canziani, N. A. Leary, D. J. Dokken and K. S. White, Climate Change 2001: Impacts, Adaption and Vulnerability Cambridge University Press, Cambridge, 2001. [36] J. Morris, S. Paltsev and J. Reilly, Marginal abatement costs and marginal welfare costs for greenhouse gas emissions reductions: Results from the EPPA model, Environmental Modeling & Assessment, 17 (2012), 325-336. Available from: https://globalchange.mit.edu/sites/default/files/MITJPSPGC_Rpt164.pdf. doi: 10.1007/s10666-011-9298-7. [37] S. C. Morris, G. A. Goldstein and V. M. Fthenakis, NEMS and MARKEL-MACRO models for energy-environmental-economic analysis: A comparison of the electricity and carbon reduction projections, Environ. Model. Assess., 7 (2002), 207-216. [38] S. Pye, K. Flecher, A. Gardiner, T. Angelini, J. Greenleaf and T. Wiley, Review and update of UK abatement costs curves for the industrial, domestic and non-domestic sectors, 2008. Available from: http://www.theccc.org.uk/wp-content/uploads/2008/12/MACC-Energy-End-Use-Final-Report-v3.2.pdf. [39] N. Rivers and M. Jaccard, Useful models for simulating policies to induce technological change, Energ. Policy, 34 (2006), 2038-2047.  doi: 10.1016/j.enpol.2005.02.003. [40] C. Schinckus, How to value GDP-linked collar bonds? An introductory perspective, Theor. Econ. Lett., 3 (2013), 152-155.  doi: 10.4236/tel.2013.33024. [41] Q. Song, Convergence of Markov chain approximation on generalized HJB equation and its applications, Automatica, 44 (2008), 761-766.  doi: 10.1016/j.automatica.2007.07.014. [42] F. Teng and S. Xu, Definition of Business as Usual and its impacts on assessment of mitigation efforts, Adv. Clim. Change Res., 3 (2012), 212-219. [43] A. Tsoularis and J. Wallace, Analysis of logistic growth models, Math. Biosci., 179 (2002), 21-55.  doi: 10.1016/S0025-5564(02)00096-2. [44] J. Wang and P. A. Forsyth, Maximal use of central differencing for Hamilton-Jacobi-Bellman PDEs in finance, SIAM J. Numer. Anal., 46 (2008), 1580-1601.  doi: 10.1137/060675186. [45] K. Wang, C. Wang and J. Chen, Analysis of the economic impact of different Chinese climate policy options based on a CGE model incorporating endogenous technological change, Energ. Policy, 37 (2009), 2930-2940. [46] X. Yang and J. Liang, Minimization of the national cost due to carbon emission, Systems Eng. Theor. Pract., 34 (2014), 640-647. [47] E. Zagheni and F. Billari, A cost valuation model based on a stochastic representation of the IPAT equation, Popul. Environ., 29 (2007), 68-82.  doi: 10.1007/s11111-008-0061-1.

show all references

##### References:
 [1] F. Ackerman and R. Bueno, Use of McKinsey abatement cost curves for climate economics modeling, 2011. Available from: http://www.sei-international.org/mediamanager/documents/Publications/Climate/sei-workingpaperus-1102.pdf. [2] F. Ackerman, E. A. Stanton and R. Bueno, CRED: A new model for climate and development, Ecol. Econ., 85 (2013), 166-176. [3] A. A. A. Almihoub, H. M. Mula and M. M. Rahman, Marginal abatement cost curves (MACCs): Important approaches to obtain (firm and sector) greenhouse gases (GHGs) reduction, Int. J. Econ. Financ., 5 (2013), 35-54. [4] R. Belauar, A. Fahim and N. Touzi, Optimal production policy under the carbon emission market, 2011. Available from: http://www.math.fsu.edu/~fahim/carbon.pdf. [5] C. Böhringer and T. F. Rutherford, Combining bottom-up and top-down, Energ. Econ., 30 (2008), 574-596. [6] R. Carmona, M. Fehr and J. Hinz, Optimal stochastic control and carbon price formation, SIAM J. Control. Optim., 48 (2009), 2168-2190.  doi: 10.1137/080736910. [7] R. Carmona, M. Fehr, J. Hinz and A. Porchet, Market design for emission trading schemes, SIAM Rev., 52 (2010), 403-452.  doi: 10.1137/080722813. [8] U. Cetin and M. Verschuere, Pricing and hedging in carbon emissions markets, Int. J. Theor. Appl. Financ., 12 (2009), 949-967.  doi: 10.1142/S0219024909005531. [9] M. Chamon and P. Mauro, Pricing growth-indexed bonds, J. Bank. Financ., 30 (2006), 3349-3366. [10] Y. Chen, Second-order Parabolic Partial Differential Equation Beijing University Press, Beijing, 2003. [11] E. Commission, EU action against climate change: The EU emissions trading scheme, 2009. Available from: http://www.ab.gov.tr/files/ardb/evt/1_avrupa_birligi/1_6_raporlar/1_3_diger/environment/eu_emmissions_trading_scheme.pdf. [12] E. Commission, The EU emissions trading system (EU ETS), 2013. Available from: https://ec.europa.eu/clima/sites/clima/files/factsheet_ets_en.pdf. [13] P. Criqui, S. Mima and L. Viguier, Marginal abatement costs of CO$_2$ emission reductions, geographical flexibility and concrete ceilings: An assessment using the POLES model, Energ. Policy, 27 (1999), 585-601. [14] T. Dietz and E. A. Rosa, Effects of population and affluence on CO$_2$ emissions, P. Natl. Acad. Sci. USA, 94 (1997), 175-179. [15] A. D. Ellerman and B. K. Buchner, The European Union Emissions Trading Scheme: Origins, allocation, and early results, REEP, 1 (2007), 66-87. [16] A. D. Ellerman and A. Decaux, Analysis of post-Kyoto CO2 emissions trading using marginal abatement curves, 1998. Available from: http://dspace.mit.edu/handle/1721.1/3608. [17] P. Enkvist, T. Nauclér and J. Rosander, A cost curve for greenhouse gas reduction, McKinsey Quarterly, 1 (2007), 35-45. [18] H. Fallman, Revision of the EU emissions trading system, Available from: http://ec.europa.eu/enterprise/sectors/chemicals/files/wg_7_8fer08/15fallmann_ets_en.pdf. [19] M. Fehr and J. Hinz, A quantitative approach to carbon price risk modeling, Available from: http://www.bbk.ac.uk/cfc/pdfs/conference\%20papers/Thurs/FehrHinz.pdf. [20] B. S. Fisher, N. Nakicenovic, K. Alfsen, J. C. Morlot, F. De La Chesnaye, J. Hourcade, K. Jiang, M. Kainuma, E. La Rovere, A. Matysek, A. Rana, K. Riahi, R. Richels, S. Rose, D. V. Vuuren and R. Warren, Issues Related to Mitigation in the Long-term Context In Climate Change 2007: Mitigation. Contribution of Working Group Ⅲ to the Fourth Assessment Report of the Inter-governmental Panel on Climate Change [B. Metz, O. R. Davidson, P. R. Bosch, R. Dave and L. A. Meyer] Cambridge University Press, Cambridge, 2007. [21] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions Springer-Verlag, New York, 1993. [22] P. A. Forsyth and G. Labahn, Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance, 2007. Available from: https://cs.uwaterloo.ca/~paforsyt/hjb.pdf. [23] C. Hepburn, Carbon trading: A review of the Kyoto mechanisms, Annu. Rev. Environ. Resour., 32 (2007), 375-393. [24] J. Hourcade, M. Jaccard, C. Bataille and F. Ghersi, Hybrid modeling: New answers to old challenges, Energy J., 2 (2006), 1-11. [25] B. Hu, Blow-up Theories for Semilinear Parabolic Equations Lecture Notes in Mathematics, 2018, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18460-4. [26] International Emissions Trading Association (IETA), The world’s carbon markets: A case study guide to emissions trading, 2013. Available from: http://www.edf.org/sites/default/files/EDF_IETA_Mexico_Case_Study_May_2013.pdf. [27] M. Jakob, Marginal costs and co-benefits of energy efficiency investments: the case of the Swiss residential sector, Energ. Policy, 34 (2006), 172-187. [28] L. Jiang, Mathematical Modeling and Methods for Option Pricing Translated from the 2003 Chinese original by Canguo Li, World Scientific Publishing Co. , Inc. , River Edge, NJ, 2005. doi: 10.1142/5855. [29] F. Kesicki, Marginal abatement cost curves for policy making--expert-based vs. model-based curves, P. IAEE Int. Conf., (2010), 1-19. [30] F. Kesicki, Marginal abatement cost curves: Combining energy system modeling and decomposition analysis, Environ. Model. Assess., 18 (2013), 27-37. [31] F. Kasicki and N. Strachan, Marginal abatement cost (MAC) curves: Confronting theory and practice, Environ. Sci. Policy, 14 (2011), 1195-1204.  doi: 10.1016/j.envsci.2011.08.004. [32] G. Klepper and S. Peterson, Marginal abatement cost curves in general equilibrium: The influence of world energy prices, Resour. Energy Econ., 28 (2006), 1-23. [33] S. Kruse, M. Meitner and M. Schröder, On the pricing of GDP-linked financial products, Appl. Financ. Econ., 15 (2005), 1125-1133. [34] G. M. Lieberman, Second Order Parabolic Differential Equations World Scientific Publishing Co. Pte. Ltd. , Singapore, 1996. doi: 10.1142/3302. [35] J. J. McCarthy, O. F. Canziani, N. A. Leary, D. J. Dokken and K. S. White, Climate Change 2001: Impacts, Adaption and Vulnerability Cambridge University Press, Cambridge, 2001. [36] J. Morris, S. Paltsev and J. Reilly, Marginal abatement costs and marginal welfare costs for greenhouse gas emissions reductions: Results from the EPPA model, Environmental Modeling & Assessment, 17 (2012), 325-336. Available from: https://globalchange.mit.edu/sites/default/files/MITJPSPGC_Rpt164.pdf. doi: 10.1007/s10666-011-9298-7. [37] S. C. Morris, G. A. Goldstein and V. M. Fthenakis, NEMS and MARKEL-MACRO models for energy-environmental-economic analysis: A comparison of the electricity and carbon reduction projections, Environ. Model. Assess., 7 (2002), 207-216. [38] S. Pye, K. Flecher, A. Gardiner, T. Angelini, J. Greenleaf and T. Wiley, Review and update of UK abatement costs curves for the industrial, domestic and non-domestic sectors, 2008. Available from: http://www.theccc.org.uk/wp-content/uploads/2008/12/MACC-Energy-End-Use-Final-Report-v3.2.pdf. [39] N. Rivers and M. Jaccard, Useful models for simulating policies to induce technological change, Energ. Policy, 34 (2006), 2038-2047.  doi: 10.1016/j.enpol.2005.02.003. [40] C. Schinckus, How to value GDP-linked collar bonds? An introductory perspective, Theor. Econ. Lett., 3 (2013), 152-155.  doi: 10.4236/tel.2013.33024. [41] Q. Song, Convergence of Markov chain approximation on generalized HJB equation and its applications, Automatica, 44 (2008), 761-766.  doi: 10.1016/j.automatica.2007.07.014. [42] F. Teng and S. Xu, Definition of Business as Usual and its impacts on assessment of mitigation efforts, Adv. Clim. Change Res., 3 (2012), 212-219. [43] A. Tsoularis and J. Wallace, Analysis of logistic growth models, Math. Biosci., 179 (2002), 21-55.  doi: 10.1016/S0025-5564(02)00096-2. [44] J. Wang and P. A. Forsyth, Maximal use of central differencing for Hamilton-Jacobi-Bellman PDEs in finance, SIAM J. Numer. Anal., 46 (2008), 1580-1601.  doi: 10.1137/060675186. [45] K. Wang, C. Wang and J. Chen, Analysis of the economic impact of different Chinese climate policy options based on a CGE model incorporating endogenous technological change, Energ. Policy, 37 (2009), 2930-2940. [46] X. Yang and J. Liang, Minimization of the national cost due to carbon emission, Systems Eng. Theor. Pract., 34 (2014), 640-647. [47] E. Zagheni and F. Billari, A cost valuation model based on a stochastic representation of the IPAT equation, Popul. Environ., 29 (2007), 68-82.  doi: 10.1007/s11111-008-0061-1.
Plots and ACFs of ln(GDP) and differentiated ln(GDP)
MAC curve for China and comparison of fitting results
Relationship between carbon emission, time and minimal cost
Relationship between $\ln(\epsilon)$ and $\ln{N}$
Relationship between carbon emission, time and optimal policy
Relationship between optimal policy and parameters
Relationship between optimal policy, policy boundary and upper bound
Relationship between minimal cost and parameters
Relationship between minimal cost and upper bound for policy
Relationship between optimal policy and technical parameters
Relationship between minimal cost and technical parameters
Fitting results: parabolic model VS CRED model
 Method m1 m2 R2 parabolic 0.0115 0.0162 0.9928 CRED 0.2142 7.6788 0.9342
 Method m1 m2 R2 parabolic 0.0115 0.0162 0.9928 CRED 0.2142 7.6788 0.9342
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