American Institute of Mathematical Sciences

Advanced
October  2007, 8(3): 569-587. doi: 10.3934/dcdsb.2007.8.569

Analysis and discretization of an optimal control problem for the forced Fisher equation

Max Gunzburger 1, , Sung-Dae Yang 1, and  Wenxiang Zhu 2,

1. 

School of Computational Science, Florida State University, Tallahassee, FL 32306-4120, United States, United States
2. 

Department of Mathematics, Idaho State University, Pocatello, ID 83209, United States

Received  September 2006 Revised  February 2007 Published  July 2007

An optimal control problem for the forced Fisher equation is considered. The control is an artificially introduced genotype and the objective is to match, as well as possible, a specified gene frequency. The existence of a solution of the optimal control problem is proved and an optimality system is derived through the Lagrange multiplier technique. Numerical approximations of the optimality system are defined using finite element methods to effect spatial discretization and a backward Euler method for the time discretiza- tion. Convergence of semi-discrete in time approximations of the state system is proved and a gradient method for solving the nonlinear discrete systems is developed. The results of some preliminary computational experiments are provided.
Keywords: Forced Fisher equation, existence of optimal controls, optimality systems, finite element methods..
Mathematics Subject Classification: Primary: 49J20, 49K20, 65M60, 92-08; Secondary: 35K20, 92B9.
Citation: Max Gunzburger, Sung-Dae Yang, Wenxiang Zhu. Analysis and discretization of an optimal control problem for the forced Fisher equation. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 569-587. doi: 10.3934/dcdsb.2007.8.569
[1]

Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927
[2]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641
[3]

Runchang Lin, Huiqing Zhu. A discontinuous Galerkin least-squares finite element method for solving Fisher's equation. Conference Publications, 2013, 2013 (special) : 489-497. doi: 10.3934/proc.2013.2013.489
[4]

Jiongmin Yong. Optimality conditions for controls of semilinear evolution systems with mixed constraints. Discrete & Continuous Dynamical Systems, 1995, 1 (3) : 371-388. doi: 10.3934/dcds.1995.1.371
[5]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077
[6]

Kim Dang Phung, Gengsheng Wang, Xu Zhang. On the existence of time optimal controls for linear evolution equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 925-941. doi: 10.3934/dcdsb.2007.8.925
[7]

Wolf-Jüergen Beyn, Janosch Rieger. Galerkin finite element methods for semilinear elliptic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 295-312. doi: 10.3934/dcdsb.2013.18.295
[8]

Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034
[9]

Qun Lin, Hehu Xie. Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods. Inverse Problems & Imaging, 2013, 7 (3) : 795-811. doi: 10.3934/ipi.2013.7.795
[10]

Zhangxin Chen. On the control volume finite element methods and their applications to multiphase flow. Networks & Heterogeneous Media, 2006, 1 (4) : 689-706. doi: 10.3934/nhm.2006.1.689
[11]

Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096
[12]

Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, 2021, 29 (3) : 2489-2516. doi: 10.3934/era.2020126
[13]

Asif Yokus, Mehmet Yavuz. Novel comparison of numerical and analytical methods for fractional Burger–Fisher equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2591-2606. doi: 10.3934/dcdss.2020258
[14]

El Hassan Zerrik, Nihale El Boukhari. Optimal bounded controls problem for bilinear systems. Evolution Equations & Control Theory, 2015, 4 (2) : 221-232. doi: 10.3934/eect.2015.4.221
[15]

Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control & Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017
[16]

Bassam Kojok. Global existence for a forced dispersive dissipative equation via the I-method. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1401-1419. doi: 10.3934/cpaa.2009.8.1401
[17]

Dongho Kim, Eun-Jae Park. Adaptive Crank-Nicolson methods with dynamic finite-element spaces for parabolic problems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 873-886. doi: 10.3934/dcdsb.2008.10.873
[18]

A. Naga, Z. Zhang. The polynomial-preserving recovery for higher order finite element methods in 2D and 3D. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 769-798. doi: 10.3934/dcdsb.2005.5.769
[19]

Tao Lin, Yanping Lin, Weiwei Sun. Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 807-823. doi: 10.3934/dcdsb.2007.7.807
[20]

Juan Wen, Yaling He, Yinnian He, Kun Wang. Stabilized finite element methods based on multiscale enrichment for Allen-Cahn and Cahn-Hilliard equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021074

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (54)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences