American Institute of Mathematical Sciences

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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
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