# American Institute of Mathematical Sciences

August  2019, 24(8): 3667-3687. doi: 10.3934/dcdsb.2018310

## Pollution control for switching diffusion models: Approximation methods and numerical results

 1 Department of Mathematics, Wayne State University, Detroit, MI 48202, USA 2 Department of Mathematics, University of Georgia, Athens, GA 30602, USA 3 Department of Electrical and Computer Engineering, Wayne State University, Detroit, MI 48202, USA

* Corresponding author: George Yin

Received  March 2018 Revised  July 2018 Published  August 2019 Early access  November 2018

Fund Project: This research was supported in part by the Army Research Office under grant W911NF-15-1-0218. The research of Q. Zhang was also supported in part by the Simons Foundation under 235179.

This work focuses on optimal pollution controls. The main effort is devoted to obtaining approximation methods for optimal pollution control. To take into consideration of random environment and other random factors, the control system is formulated as a controlled switching diffusion. Markov chain approximation techniques are used to design the computational schemes. Convergence of the algorithms are obtained. To demonstrate, numerical experimental results are presented. A particular feature is that computation using real data sets is provided.

Citation: Caojin Zhang, George Yin, Qing Zhang, Le Yi Wang. Pollution control for switching diffusion models: Approximation methods and numerical results. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3667-3687. doi: 10.3934/dcdsb.2018310
##### References:
 [1] G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations, ESAIM: Mathematical Modelling and Numerical Analysis, 36 (2002), 33-54.  doi: 10.1051/m2an:2002002. [2] S. De Vito, E. Massera, M. Piga, L. Martinotto and G. Di Francia, On field calibration of an electronic nose for benzene estimation in an urban pollution monitoring scenario, Sensors and Actuators B: Chemical, 129 (2008), 750-757. [3] X. Han and P. E. Kloeden, Random Ordinary Differential Equations and Their Numerical Solution, Springer, Singapore, 2017. doi: 10.1007/978-981-10-6265-0. [4] H. Jasso-Fuentes and G. Yin, Advanced Criteria for controlled Markov-Modulated Diffusions in an Infinite Horizon: Overtaking, Bias, and Blackwell Optimality, Science Press, Beijing, China, 2013. [5] K. Kawaguchi and H. Morimoto, Long-run average welfare in a pollution accumulation model, J. Econom. Dyn. Control, 31 (2007), 703-720.  doi: 10.1016/j.jedc.2006.04.001. [6] E. Keeler, M. Spence and R. Zeckhauser, The optimal control of pollution, J. Economic Theory, 4 (1972), 19-34. [7] K. Khalvati, S. A. Park, J. Dreher and R. Rao, A probabilistic model of social decision making based on reward maximization, Advances in Neural Information Processing Systems, (2016), 2901-2909. [8] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5. [9] H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, MIT Press, Cambridge, MA, 1984. [10] H. J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Springer Science & Business Media, 1992. doi: 10.1007/978-1-4684-0441-8. [11] D. Larose, K-nearest neighbor algorithm, Discovering Knowledge in Data: An Introduction to Data Mining, (2005), 90-106. [12] X. Lu, G. Yin, Q. Zhang, C. Zhang and X. Guo, Building up an illiquid stock position subject to expected fund availability: Optimal controls and numerical methods, Appl. Math. Optim., 76 (2017), 501-533.  doi: 10.1007/s00245-016-9359-z. [13] Q. Song, G. Yin and Z. Zhang, Numerical method for controlled regime-switching diffusions and regime-switching jump diffusions, Automatica, 42 (2006), 1147-1157.  doi: 10.1016/j.automatica.2006.03.016. [14] G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications: A Two-Time-Scale Approach, Second Edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4346-9. [15] G. Yin, Q. Zhang and G. Badowski, Discrete-time singularly perturbed Markov chains: Aggregation, occupation measures, and switching diffusion limit, Adv. in Appl. Probab., 35 (2003), 449-476.  doi: 10.1239/aap/1051201656. [16] G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6. [17] Q. Zhang and G. Yin, On nearly optimal controls of hybrid LQG problems, IEEE Trans. Automat. Control, 44 (1999), 2271-2282.  doi: 10.1109/9.811209. [18] Q. Zhang, C. Zhang and G. Yin, Optimal stopping of two-time scale Markovian systems: Analysis, numerical methods, and applications, Nonlinear Analysis: Hybrid Systems, 26 (2017), 151-167.  doi: 10.1016/j.nahs.2017.05.005.

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##### References:
 [1] G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations, ESAIM: Mathematical Modelling and Numerical Analysis, 36 (2002), 33-54.  doi: 10.1051/m2an:2002002. [2] S. De Vito, E. Massera, M. Piga, L. Martinotto and G. Di Francia, On field calibration of an electronic nose for benzene estimation in an urban pollution monitoring scenario, Sensors and Actuators B: Chemical, 129 (2008), 750-757. [3] X. Han and P. E. Kloeden, Random Ordinary Differential Equations and Their Numerical Solution, Springer, Singapore, 2017. doi: 10.1007/978-981-10-6265-0. [4] H. Jasso-Fuentes and G. Yin, Advanced Criteria for controlled Markov-Modulated Diffusions in an Infinite Horizon: Overtaking, Bias, and Blackwell Optimality, Science Press, Beijing, China, 2013. [5] K. Kawaguchi and H. Morimoto, Long-run average welfare in a pollution accumulation model, J. Econom. Dyn. Control, 31 (2007), 703-720.  doi: 10.1016/j.jedc.2006.04.001. [6] E. Keeler, M. Spence and R. Zeckhauser, The optimal control of pollution, J. Economic Theory, 4 (1972), 19-34. [7] K. Khalvati, S. A. Park, J. Dreher and R. Rao, A probabilistic model of social decision making based on reward maximization, Advances in Neural Information Processing Systems, (2016), 2901-2909. [8] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5. [9] H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, MIT Press, Cambridge, MA, 1984. [10] H. J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Springer Science & Business Media, 1992. doi: 10.1007/978-1-4684-0441-8. [11] D. Larose, K-nearest neighbor algorithm, Discovering Knowledge in Data: An Introduction to Data Mining, (2005), 90-106. [12] X. Lu, G. Yin, Q. Zhang, C. Zhang and X. Guo, Building up an illiquid stock position subject to expected fund availability: Optimal controls and numerical methods, Appl. Math. Optim., 76 (2017), 501-533.  doi: 10.1007/s00245-016-9359-z. [13] Q. Song, G. Yin and Z. Zhang, Numerical method for controlled regime-switching diffusions and regime-switching jump diffusions, Automatica, 42 (2006), 1147-1157.  doi: 10.1016/j.automatica.2006.03.016. [14] G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications: A Two-Time-Scale Approach, Second Edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4346-9. [15] G. Yin, Q. Zhang and G. Badowski, Discrete-time singularly perturbed Markov chains: Aggregation, occupation measures, and switching diffusion limit, Adv. in Appl. Probab., 35 (2003), 449-476.  doi: 10.1239/aap/1051201656. [16] G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6. [17] Q. Zhang and G. Yin, On nearly optimal controls of hybrid LQG problems, IEEE Trans. Automat. Control, 44 (1999), 2271-2282.  doi: 10.1109/9.811209. [18] Q. Zhang, C. Zhang and G. Yin, Optimal stopping of two-time scale Markovian systems: Analysis, numerical methods, and applications, Nonlinear Analysis: Hybrid Systems, 26 (2017), 151-167.  doi: 10.1016/j.nahs.2017.05.005.
The control actions in two states
The value functions in two states
The control actions and value functions in two states
The histograms of distributions
Value functions and optimal controls for NO2
Control actions of NO2 on testing set based on optimal strategy
Control actions of NOx on testing set based on optimal strategy
Value functions and optimal controls for 2 dimension system
Value functions and optimal controls for a 4 dimension case
Control actions on testing set based on optimal strategy
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