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.
Table of Contents
1 Introduction
2 Optimality conditions for optimal control problems governed by a finite number of integral equations
2.1 A class of optimal control problems
2.2 Regularity conditions and Lagrange multiplier rule
2.3 Necessary and sufficient optimality conditions for the optimal control problem
3 The conditioned gradient method for linear-quadratic control problems
3.1 Dicretization and numerical implementation
3.2 Computation of the state equation
3.3 Computation of the adjoint equation
3.4 Determination of the direction of descent and computation of the cost function
4 Numerical example
Research Objectives and Themes
This research aims to develop a novel numerical approximation method for solving optimal control problems governed by weakly singular Volterra integral equations. By utilizing a conjugate gradient method combined with a collocation approach on graded mesh points, the study seeks to minimize objective functions subject to state equation constraints and control limitations.
- Theoretical derivation of necessary and sufficient optimality conditions.
- Application of the "Two Norm Technique" to address basic space obstacles.
- Implementation of the conditioned gradient method for linear-quadratic control.
- Discretization of state and adjoint equations using collocation on graded meshes.
- Numerical verification of efficiency through test examples.
Excerpt from the Book
3.1 Dicretization and numerical implementation
The state equations (17) and the adjoint equation (19) both describe Volterra integral equations. Thence for their computation we will apply the collocation method described in section 2.3 of chapter 2. The time interval [0, T] will be discretized into a finite number of subintervals as indicated in Figure 1.
[0, T] = union_{j=0}^{Noint-1} [Tj, Tj+1),
with Noint = number of (equidistant) subintervals. Tj is obtained as follows
Tj = T/Noint * j, for j = 0, ..., Noint.
The control function, u(t), describes a step function and it is constant in each subinterval. Thus uj+1 in [Tj, Tj+1). This discretization is in conformity with the conditions of Theorem 2.3 and Theorem 2.4, which state that the convergence of the method applied can be guaranteed only when the functions under the integral signs are continuous in the required interval. In order to abide by this convergence rule we consider the subintervals as "independent" and separate entities, and in the same way we compute the state equation (17) and the adjoint equation (19) in these intervals. Each subinterval [Tj, Tj+1) would be discretized further along the lines of section 2.3 to obtain the discrete points of t. The discrete points {tk} of an independent subinterval in the forward integration (17) would read
Summary of Chapters
1 Introduction: Provides an overview of optimal control problems in science and engineering and introduces the research scope.
2 Optimality conditions for optimal control problems governed by a finite number of integral equations: Develops the theoretical foundation, including regularity conditions and the Lagrange multiplier rule.
3 The conditioned gradient method for linear-quadratic control problems: Describes the algorithmic implementation of the numerical solver and the discretization strategy.
4 Numerical example: Demonstrates the practical applicability and efficiency of the proposed method using a specific test case.
Keywords
Optimal control problem, Volterra integral equation, Collocation method, Conjugate gradient method, Numerical method, Weakly singular kernel, Optimality conditions, Graded mesh, Discretization, Adjoint equation, Banach spaces, Lagrange multiplier, Linear-quadratic, Convergence, Step function.
Frequently Asked Questions
What is the primary focus of this research?
The work focuses on approximating solutions for optimal control problems where the underlying state equations are weakly singular Volterra integral equations.
What is the main objective of the proposed method?
The goal is to effectively minimize a defined objective function subject to state equation constraints and control restrictions using a new numerical approach.
Which numerical technique is employed for optimization?
The research employs the conjugate gradient method paired with a collocation approach for the discretization of the problem.
How are the state and adjoint equations handled?
They are handled via discretization over graded mesh points, treating subintervals as independent entities to ensure convergence.
What theoretical tool addresses the differentiability obstacles?
The "Two Norm Technique" is applied to navigate issues related to the basic spaces where integral operators are defined.
What characterizes the control function in this study?
The control function is modeled as a step function that remains constant within each specific subinterval.
How is the discretization grid constructed?
The time interval [0, T] is partitioned into a finite number of subintervals, and each subinterval is further discretized to facilitate numerical integration.
What specific result was achieved in the numerical example?
The numerical example demonstrated the method's ability to minimize the cost functional to nearly zero (0.000233) after eight iterations.
- Citation du texte
- Henry Ekah-Kunde (Auteur), 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.grin.com/document/371529
