A novel approach of numerical optimization for control theory problems based on generalization of Gigena's method

Abstract

In this article, a new efficient numerical method is proposed to solve control theory problems in a finite difference setting. The method was originally devised by Gigena for determining and analyzing constrained local extrema without using Lagrange multipliers with the use of differential forms. He efficiently solved a number of algebraic problems and proved his method to be a better alternative as compared with already existing methods. Inspired by his work, we extended the scope of his work and generalized his technique to solve optimal control problems. The results are highly encouraging, and it can be seen that the proposed method has great advantages in terms of efficiency and computational cost in solving optimal control problems. This new approach provides a new way of looking at solving numerous other optimal control problems, such as discrete time models, with ease.