Mixed Implicit Galerkin â€“ Frank Wolf, Gradient and Gradient Projection Methods for Solving Classical Optimal Control Problem Governed by Variable Coefficients, Linear Hyperbolic, Boundary Value Problem

Eman H Mukhalef Al-Rawdanee,Jamil A Ali Al-Hawasy

doi:10.24996/ijs.2020.61.9.17

Abstract

This paper deals with testing a numerical solution for the discrete classical optimal control problem governed by a linear hyperbolic boundary value problem with variable coefficients. When the discrete classical control is fixed, the proof of the existence and uniqueness theorem for the discrete solution of the discrete weak form is achieved. The existence theorem for the discrete classical optimal control and the necessary conditions for optimality of the problem are proved under suitable assumptions. The discrete classical optimal control problem (DCOCP) is solved by using the mixed Galerkin finite element method to find the solution of the discrete weak form (discrete state). Also, it is used to find the solution for the discrete adjoint weak form (discrete adjoint) with the Gradient Projection method (GPM) , the Gradient method (GM), or the Frank Wolfe method (FWM) to the DCOCP. Within each of these three methods, the Armijo step option (ARSO) or the optimal step option (OPSO) is used to improve (to accelerate the step) the solution of the discrete classical control problem. Finally, some illustrative numerical examples for the considered discrete control problem are provided. The results show that the GPM with ARSO method is better than GM or FWM with ARSO methods. On the other hand, the results show that the GPM and GM with OPSO methods are better than the FWM with the OPSO method.

Highlights

Optimal control problems (OCP) have various applications [1, 2]
This paper investigates the numerical solution of the DCCOCP that is described by the LHBVPVC
The proof of the existence and uniqueness theorem for the discrete solution (DS) of the discrete weak form (DWF) for the LHBVPVC is achieved

Summary

Introduction

Optimal control problems (OCP) have various applications [1, 2]. These problems are usually governed by partial differential equations (PDEs) or ordinary differential equations (ODEs). Many researchers have been interested to study the numerical solution of optimal control problems described by nonlinear elliptic PDEs [3, 4], by semilinear parabolic PDEs [5,6,7], or by one dimensional linear hyperbolic PDEs with constant coefficients (LHPDES) [8]. The researchers include two dimensional linear and nonlinear hyperbolic PDEs with constant coefficients [9,10,11,12], or by nonlinear ODEs [13] These works attracted our attention to focus our interest on studying OCP described by LHPDES but with variable coefficients (LHBVPVC). The continuous classical optimal control problem (CCOCP) is described, which is discretized by applying the Galerkin finite element method (GFEM). Some illustrative examples for this considered problem are given to show the accuracy and the efficiency of each of the three methods

The Statement of the CCOCP

Theorem

Corollary

ALGORITHIM

Application 1: Consider the following CCOCP governed by the LHBVPVC

Conclusions