Conjugate gradient method for the solution of optimal control problems governed by weakly singular Volterra integral equations with the use of the collocation method

Abstract

In this research, a novel method to approximate the solution of optimal con- trol problems governed by Volterra integral equations of weakly singular types is proposed. The method introduced here is the conjugate gradient method with a discretization of the problem based on the collocation approach on graded mesh points for non linear Volterra integral equations with singular kernels. Neces- sary and sufficient optimality conditions for optimal control problems are also discussed. Some examples are presented to demonstrate the efficiency of the method.

Keywords: Optimal control problem; Volterra integral equation; Collocation method; Conjugate gradient method; Numerical method.

1 Introduction

Optimal control problems appear in a wide range of applications in engineering and science. Many problems in epidemiology, biology, economics and the like belong to the class of optimal control problems governed by (Volterra) integral equations.

In the last couple of decades advances in the solution of optimal control prob- lems have been made. Different types of methods have been proposed to solve optimal control problems of this class.

A general model of an optimal control problem of this type involves the mini- mization of an objective function governed by an integral equation, which con- tains and depends on control functions that are selected within certain limits. The objective here is to solve for the control that satisfies the state equation and its restrictions, and with the aid of this control as well as the desired state, the solution of the integral equation, minimize the objective function.

Our control is one that is governed by Volterra integral equation with weakly singular kernel. A treatment of the solution of Volterra integral equations has been handled in several articles, for instance in the works of Brunner [7], Brunner and van der Houwen [11] and Te Riele [22].

A theoretical analysis of the optimal control problem involves the investigation of the existence of a solution to it, that is, the treatment of necessary and suf- ficient optimality conditions. In the handling of these conditions problems will be encountered with respect to the basic space. The integral operators occur- ring in this problem are differentiable only in the L∞ space, while the sufficient optimality conditions can only be satisfied in the Lp (generally p = 2) spaces. To overcome this obstacle we shall apply the "Two Norm Technique", which has been perfectly handled in the works of Maurer [19] and Tröltzsch [23] and [24]. We shall discuss briefly on the necessary optimality conditions as well as the regularity conditions, which are prerequisites for optimality and the Lagrange multiplier rule. Sufficient optimality conditions have been treated in several articles such as the publications of Maurer [19], Maurer and Zowe [20], Goldberg and Tröltzsch [15], Casa, Tröltzsch and Unger [12] and Tröltzsch [23], [24] and [25]. The approach used by Casas, Tröltzsch and Unger [12] for nonlinear elliptical boundary control problems shall be applied to our problem. These theoretical investigations provide a basis for a numerical treatment of the optimal control problems, which is the subject matter of this work. The numerical method chosen for solving optimal control problems here is the con- jugate gradient method.

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