On the Discretized Algorithm for Optimal Problems Constrained by Differential Equation with Real Coefficients

O Olotu

doi:10.3844/jmssp.2007.237.242

Abstract

A discretized scheme, Discretized Continuous Algorithm (DCA), for solving constrained quadratic optimal control problems was developed to ease the computational cumbersomeness inherent in some existing algorithms, particularly, the Function Space A lgorithm (FSA) by replacing the integral by a series of summation. In order to accomplish this numerical scheme, we resort to a finite approximation of it by discretizing its time interval and using finite difference method for its differential constraint. Using the penalty function method, an unconstrained formulation of the problem was obtained. With the bilinear form expression of the problem, an associated operator was constructed which aided the scheme for the solution of such class of problems. A sample problem was examined to test the effectiveness of the scheme as to convergence with relation to other existing schemes such as Extended Conjugate Gradient Method (ECGM), Multiplier Imbedding Extended Conjugate Gradient Method (MECGM) and Function Space Algorithm (FSA) for solving penalized functional of optimal control problem characterized by non-linear integral quadratic nature.

Highlights

The discretized scheme, Discretized Continuous Algorithm (DCA), with less computational rigour was proposed and compared to some existing algorithms, the Function Space A lgorithm (FSA) which circumvented the use of operator, for solving a class of quadratic optimal control problems
The discretized scheme, DCA, with less computational rigour was proposed and compared to some existing algorithms, the FSA which circumvented the use of operator, for solving a class of quadratic optimal control problems
The problem has been solved by other numerical methods such as Function Space Algorithm(FSA), Extended Conjugate Gradient Method (ECGM) and Multiplier Imbedding Extended Conjugate Gradient Method(MECGM)[ ] with results tabulated below

Summary

INTRODUCTION

The discretized scheme, DCA, with less computational rigour was proposed and compared to some existing algorithms, the FSA which circumvented the use of operator, for solving a class of quadratic optimal control problems. ECGM and MECGM based on[1] on function minimization reviewed by[4] were ingredients to the development of the discretized scheme. A generalized constrained formulation of the problem is given below for the discretization exercise of the scheme

MATERIALS AND METHODS

COMMENTS AND CONCLUSION

Objective function