Analytic-Approximate Solution for a Class of Nonlinear Optimal Control Problems by Homotopy Analysis Method

Abstract

In this paper, we apply the homotopy analysis method (HAM) for nonlinear quadratic optimal control problems (OCP's). This method is employed to solve extreme conditions obtained from the Pontryagin's maximum principle (PMP). Applying the HAM, we in essence transfer a nonlinear two-point boundary value problem (TPBVP), into an infinite number of linear sub-problems, and then use the sum of the solutions of its first several sub-problems to approximate the exact solution. Note that such a kind of transformation does not need the existence of any small or large parameters in governing equation and initial/boundary conditions. The comparison of the HAM results with the Measure theory method, Modal series and collocation method solutions are made. Some illustrative examples are given to demonstrate the simplicity and efficiency of the proposed method.

A Laguerre spectral method for quadratic optimal control of nonlinear systems in a semi-infinite interval

“…In this paper, we use the Laguerre pseudo-spectral method to solve Equations (22)- (24). The pseudospectral derivative D N (z) of a continuous function z is defined by…”

“…The HAM [19,20] was first proposed by Liao in 1992 to solve lots of nonlinear problems. This method has been successfully applied to many nonlinear problems, such as physical models with an infinite number of singularities [21], nonlinear eigenvalue problems [22], fractional Sturm-Liouville problems [23], optimal control problems [24,25], Cahn-Hilliard initial value problem [26], semi-linear elliptic boundary value problems [27] and so on [28]. The HAM contains a certain auxiliary parameter which provides us with a simple way to adjust and control the convergence region and rate of convergence of the series solution.…”

Homotopy Perturbation Method for Brachistochrone Problem

“…In this method the solution is considered as the sum of an infinite series which usually convergence rapidly to an accurate solutions. Many researchers have been successfully applied Homotopy perturbation method to various nonlinear problems in science and engineering, such as the viscous flows of non-Newtonian fluids [1], the KdV-type equations [2], finance problems [3], nonlinear optimal control problems [4][5] and some others.…”

On the Homotopy Analysis Method and Optimal Value of the Convergence Control Parameter: Solution of Euler-Lagrange Equation

“…Since Liao [10] for the homotopy analysis method was published in 2003, more and more researchers have been successfully applying this method to various nonlinear problems in science and engineering, such as the viscous flows of non-Newtonian fluids [14], the KdV-type equations [15], finance problems [16], nonlinear optimal control problems [17] and so on.…”