Optimal Control of the Fisher Equation: A Comparison of Steepest Descent and Quasi-Newton Methods

Abstract

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.