Convergence Ratio Profile for Optimal Control Problems Governed by Ordinary Differential Equations with Matrix Coefficients

O Olotu,S.A Olorunsola

doi:10.3844/ajassp.2008.89.92

Abstract

The geometric convergence ratio, the main focus of a discretized scheme for constrained quadratic control problem was examined. In order to allow for the numerical applications of the developed scheme, discretizing the time interval and using Euler’s scheme for its differential constraint obtained a finite dimensional approximation. Applying the penalty function method, an unconstrained problem was obtained on function minimization with bilinear form expression. This finally led to the construction of an operator. The Scheme was applied to a sampled problem and it exhibited geometric convergence ratio, α, in the open interval (0, 1) as depicted in column 6 of Table 1.

Highlights

In[1,2], the scheme establishing the solution of optimal control problems constrained by evolution equation of the delay type with matrix coefficients was presented, without addressing the geometric convergence ratio profile
A class of optimal control problems constrained by ordinary differential equation with matrix coefficients is considered
Discretization of the generalized problem is obtained by discretizing its objective function and using[3] for its differential constraint

Summary

INTRODUCTION

In[1,2], the scheme establishing the solution of optimal control problems constrained by evolution equation of the delay type with matrix coefficients was presented, without addressing the geometric convergence ratio profile. A class of optimal control problems constrained by ordinary differential equation with matrix coefficients is considered. Using[4], a penalty function method is applied to convert the constrained problem into an unconstrained formulation problem. With this formulation, an associated control operator was constructed as in[2]. Where, x(t), u(t)εRn , x(t)T , u(t)T denote the transposes of x(t) and u(t) respectively. A and C are n by n and n by m matrices not necessarily symmetric respectively

Discretization

Theorem 1

Remark 1

RESULTS

DISCUSSION