In this research, a novel method to approximate the solution of optimal control 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. Necessary and sufficient optimality conditions for optimal control problems are also discussed. Some examples are presented to demonstrate the efficiency of the method.
Inhaltsverzeichnis (Table of Contents)
- Introduction
- Optimality conditions for optimal control problems governed by a finite number of integral equations
- A class of optimal control problems
- Regularity conditions and Lagrange multiplier rule
- Necessary and sufficient optimality conditions for the optimal control problem
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
The objective of this research is to develop a novel method for approximating solutions to optimal control problems governed by weakly singular Volterra integral equations. This involves using the conjugate gradient method with a collocation approach on graded mesh points. The research also explores necessary and sufficient optimality conditions for these types of problems.
- Optimal control problems governed by Volterra integral equations
- Weakly singular kernels in Volterra integral equations
- Conjugate gradient method for numerical solution
- Collocation method on graded mesh points
- Necessary and sufficient optimality conditions
Zusammenfassung der Kapitel (Chapter Summaries)
Introduction: This chapter introduces the concept of optimal control problems and their wide range of applications in various fields like engineering, epidemiology, biology, and economics. It highlights the challenges associated with solving optimal control problems governed by Volterra integral equations, particularly those with weakly singular kernels. The chapter sets the stage for the research by emphasizing the need for efficient numerical methods and discussing existing approaches for solving Volterra integral equations.
Optimality conditions for optimal control problems governed by a finite number of integral equations: This chapter delves into the theoretical foundation for solving the optimal control problem. It begins by defining a class of optimal control problems governed by a system of integral equations, introducing crucial assumptions regarding the continuity and differentiability of involved functions and the weakly singular nature of the kernel. The chapter then defines regularity conditions and the Lagrange multiplier rule, essential for establishing optimality. The concepts of the Lagrange functional, adjoint equation, and variational inequality are explained. Finally, the chapter lays out the necessary and sufficient optimality conditions, which provide a theoretical basis for the numerical method employed in the research.
Schlüsselwörter (Keywords)
Optimal control problem, Volterra integral equation, weakly singular kernel, conjugate gradient method, collocation method, numerical method, optimality conditions, Lagrange multiplier, adjoint equation.
Frequently Asked Questions: A Comprehensive Language Preview of Optimal Control Problems
What is the main topic of this research?
This research focuses on developing a novel numerical method for solving optimal control problems governed by weakly singular Volterra integral equations. It uses a conjugate gradient method with a collocation approach on graded mesh points.
What are the key themes explored in this research?
Key themes include: optimal control problems governed by Volterra integral equations; weakly singular kernels in Volterra integral equations; the conjugate gradient method for numerical solutions; collocation methods on graded mesh points; and necessary and sufficient optimality conditions.
What are the objectives of this research?
The objective is to create a new method for approximating solutions to optimal control problems with weakly singular Volterra integral equations. This involves applying the conjugate gradient method with a collocation approach on a graded mesh. The research also investigates the necessary and sufficient optimality conditions for these problems.
What methods are used in this research?
The research utilizes the conjugate gradient method and a collocation method on graded mesh points to numerically solve the optimal control problems. The theoretical foundation relies on establishing necessary and sufficient optimality conditions.
What types of equations are considered?
The research centers on optimal control problems governed by Volterra integral equations, specifically those with weakly singular kernels.
What are the chapter summaries?
The introduction provides background on optimal control problems and their applications, highlighting challenges associated with Volterra integral equations. The main chapter delves into the theoretical framework, defining the problem class, regularity conditions, the Lagrange multiplier rule, and deriving necessary and sufficient optimality conditions.
What are the necessary and sufficient optimality conditions?
The research establishes necessary and sufficient optimality conditions for the optimal control problems. These conditions, derived using the Lagrange multiplier rule and variational inequalities, provide a theoretical basis for the numerical methods employed.
What are the key words associated with this research?
Key words include: Optimal control problem, Volterra integral equation, weakly singular kernel, conjugate gradient method, collocation method, numerical method, optimality conditions, Lagrange multiplier, adjoint equation.
What is the Table of Contents?
The Table of Contents includes an Introduction and a chapter on Optimality conditions for optimal control problems governed by a finite number of integral equations, which is further divided into sections on a class of optimal control problems, regularity conditions and the Lagrange multiplier rule, and necessary and sufficient optimality conditions for the optimal control problem.
- Quote paper
- Henry Ekah-Kunde (Author), 2015, Conjugate gradient method for the solution of optimal control problems governed by weakly singular Volterra integral equations with the use of the collocation method, Munich, GRIN Verlag, https://www.hausarbeiten.de/document/371529
