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

Caojin Zhang; George Yin; Qing Zhang; Le Yi Wang

doi:10.3934/dcdsb.2018310

Article Contents

2019, Volume 24, Issue 8: 3667-3687. Doi: 10.3934/dcdsb.2018310

This issue Previous Article Invariance principle in the singular perturbations limit Next Article Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient

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
* Corresponding author: George Yin

Received: February 28, 2018

Revised: June 30, 2018

Early access: October 2018

Published: July 2019

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 65C20, 65C30, 60H35, 93E20.

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Figure 1. The control actions in two states

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Figure 2. The value functions in two states

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Figure 3. The control actions and value functions in two states

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Figure 4. The histograms of distributions

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Figure 5. Value functions and optimal controls for NO₂

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Figure 6. Control actions of NO₂ on testing set based on optimal strategy

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Figure 7. Control actions of NOx on testing set based on optimal strategy

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Figure 8. Value functions and optimal controls for 2 dimension system

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Figure 9. Value functions and optimal controls for a 4 dimension case

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Figure 10. Control actions on testing set based on optimal strategy

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