Optimal Control of the Fisher Equation: A Comparison of Steepest Descent and Quasi-Newton Methods
Resumen
This article examines the performance of the steepest descent and quasi-Newton methods, both incorporating line search strategies based on the Armijo rule and Wolfe conditions, in solving an optimal control problem governed by the Fisher equation. To handle inequality constraints, a penalty method is employed to reformulate them as equalities. The existence of solutions is analyzed, and the optimal control is characterized. Numerical experiments are conducted to compare the two methods, highlighting their features, advantages, and limitations.
Citas
[1] Haim Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2010.
[2] J. C. De los Reyes. Numerical PDE-Constrained Optimization. SpringerBriefs in Optimization. Springer International Publishing, 2015.
[3] John E. Dennis and Robert B. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia, 1996.
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[8] Suzanne Lenhart and J. A. Montero. Optimal control of harvesting in a parabolic system modeling two subpopulations. Mathematical Models and Methods in Applied Sciences, 11:1129–1141, 2001.
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[11] Julien Martin, Matthew Richardson, Davina Passeri, Nicholas Enwright, Simeon Yurek, James Flocks, Mitchell Eaton, Sara Zeigler, Hadi Charkhgard, Bradley Udell, and Elise Irwin. Decision science as a framework for combining geomorphological and ecological modeling for the management of coastal systems. Ecology and Society, 28, 03 2023.
[12] Pedro Merino and Cristhian Núñez. Numerical simulation of free boundary biviscous fluids. Bulletin of Computational Applied Mathematics, 12(2):xx–xx, 2024. Special Issue V JEM 2023.
[13] Rachael Miller Neilan and Suzanne Lenhart. An introduction to optimal control with an application in disease modeling. DIMACS Series in Discrete Mathematics, 75, 10 2010.
[14] Ibrahim Mohd Asrul Hery, Mamat Mustafa, Ghazali Puspa Liza, and Salleh Zabidin. The scaling of hybrid method in solving unconstrained optimization method. Far East Journal of Mathematical Sciences (FJMS), 2015.
[15] Michael Neubert. Marine reserves and optimal harvesting. Ecology Letters, 6:843–849, 2003.
[16] Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer, 1999.
[17] Jorge Nocedal and Stephen J. Wright. Line search methods. In Numerical Optimization, pages 30–65. Springer, 2006.
[18] Cristhian Núñez. Reducción de un modelo de control óptimo de dinámica poblacional mediante técnicas pod, 2015. Repositorio Digital EPN, Tesis Matemáticas (MAT).
[19] Tu Peng, Xu Yang, Zi Xu, and Yu Liang. Constructing an environmental friendly low-carbon-emission intelligent transportation system based on big data and machine learning methods. Sustainability, 12(19):8118, 2020.
[20] Alfio Quarteroni, Riccardo Sacco, and Fausto Saleri. Numerical Mathematics (Texts in Applied Mathematics). Springer-Verlag, Berlin, Heidelberg, 2006.
[21] Walter Rudin. Principles of Mathematical Analysis. McGraw-Hill, New York, 3 edition, 1976.
[2] J. C. De los Reyes. Numerical PDE-Constrained Optimization. SpringerBriefs in Optimization. Springer International Publishing, 2015.
[3] John E. Dennis and Robert B. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia, 1996.
[4] Lawrence C. Evans. Partial Differential Equations, volume 19. American Mathematical Society, 2022.
[5] M. Gunzburger, L. Hou, and W. Zhu. Modeling and analysis of the forced fisher equation. Nonlinear Analysis: Theory, Methods and Applications, 62:19–40, 2005.
[6] Innyoung Kim and Donghyun You. Fluid dynamic control and optimization using deep reinforcement learning. JMST Advances, 2024.
[7] John R. King and Philip M. McCabe. On the fisher–kpp equation with fast nonlinear diffusion. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 459(2038):2529–2546, 2003.
[8] Suzanne Lenhart and J. A. Montero. Optimal control of harvesting in a parabolic system modeling two subpopulations. Mathematical Models and Methods in Applied Sciences, 11:1129–1141, 2001.
[9] Suzanne M. Lenhart and Mahadev G. Bhat. Application of distributed parameter control model in wildlife damage management. Mathematical Models and Methods in Applied Sciences, 2(04):423–439, 1992.
[10] Ming Li and Qing Xie. Optimal control for a stationary population. Journal of Mathematics Research, 4, 2012.
[11] Julien Martin, Matthew Richardson, Davina Passeri, Nicholas Enwright, Simeon Yurek, James Flocks, Mitchell Eaton, Sara Zeigler, Hadi Charkhgard, Bradley Udell, and Elise Irwin. Decision science as a framework for combining geomorphological and ecological modeling for the management of coastal systems. Ecology and Society, 28, 03 2023.
[12] Pedro Merino and Cristhian Núñez. Numerical simulation of free boundary biviscous fluids. Bulletin of Computational Applied Mathematics, 12(2):xx–xx, 2024. Special Issue V JEM 2023.
[13] Rachael Miller Neilan and Suzanne Lenhart. An introduction to optimal control with an application in disease modeling. DIMACS Series in Discrete Mathematics, 75, 10 2010.
[14] Ibrahim Mohd Asrul Hery, Mamat Mustafa, Ghazali Puspa Liza, and Salleh Zabidin. The scaling of hybrid method in solving unconstrained optimization method. Far East Journal of Mathematical Sciences (FJMS), 2015.
[15] Michael Neubert. Marine reserves and optimal harvesting. Ecology Letters, 6:843–849, 2003.
[16] Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer, 1999.
[17] Jorge Nocedal and Stephen J. Wright. Line search methods. In Numerical Optimization, pages 30–65. Springer, 2006.
[18] Cristhian Núñez. Reducción de un modelo de control óptimo de dinámica poblacional mediante técnicas pod, 2015. Repositorio Digital EPN, Tesis Matemáticas (MAT).
[19] Tu Peng, Xu Yang, Zi Xu, and Yu Liang. Constructing an environmental friendly low-carbon-emission intelligent transportation system based on big data and machine learning methods. Sustainability, 12(19):8118, 2020.
[20] Alfio Quarteroni, Riccardo Sacco, and Fausto Saleri. Numerical Mathematics (Texts in Applied Mathematics). Springer-Verlag, Berlin, Heidelberg, 2006.
[21] Walter Rudin. Principles of Mathematical Analysis. McGraw-Hill, New York, 3 edition, 1976.
