DIRECT AND INDIRECT MULTIPLE SHOOTING METHODS FOR NONLINEAR OPTIMAL CONTROL PROBLEM

Abstract

In this project, we discussed nonlinear optimal control problems. Nonlinear optimal control problems are challenging to solve due to the prevalence of local minimum that prevent convergence or optimality. This project was concerned with two numerical methods (indirect and direct methods). Indirect method based on Pontryagin’s Maximum Principle optimize in an infinite dimensional function space and derive the necessary conditions for optimality using co-state variables and convert the optimal control problem into a two point boundary value problem, then solving the two point boundary value problem numerically. Direct methods were based on a discretization of the infinite dimensional optimal control problem into a finite dimensional nonlinear optimization problem. This study was described nearest neighbors optimal control. The presented concepts and methods were verified by means of different examples, where by theoretical results are numerically confirmed. We choose the test problems, so that the simple shooting method becomes unstable and a genuine multiple shooting techniques were required. By comparisons of both methods we obtained that indirect multiple shooting methods is more accurate than direct multiple shooting method. Most computations were performed using MATLAB codes. This project can be extending to solve nonlinear optimal control problems with inequality constraints using these two methods.