A conjugate gradient method-based algorithm for optimizing the generalized 2-dimensional optimal proportional control problems

Abstract

A generalized two-dimensional optimal proportional control problem constrained by ordinary differential equations is presented in this study. The analytical method and numerical technique for solving optimal proportional control problems with equality constraints are discussed. Complete discretization approach is adopted for the numerical optimization of the problem. Simpson’s ?? Rule and Fifth-order Adams Moulton Technique are employed for the discretization of the objective functional and the constraint, respectively. Quadratic penalty function method is used to transform the constrained discretized optimal proportional control problem to unconstrained problem. Numerical solution of the unconstrained optimization problem is obtained using Conjugate gradient method algorithm. Two numerical optimal proportional control problems are considered and their analytical and numerical solutions are presented. The accuracy and efficiency of the new scheme over the existing algorithms is demonstrated through the analysis of convergence. The results show that the new scheme exhibits a superlinear convergence, which is an improvement on the existing schemes.

Keywords: Proportional Control, Discretization, Quadratic Penalty Function, Convergence, Conjugate Gradient Method.