Inexact restoration and adaptive mesh refinement for constrained optimal control
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(Published version)
Date
2012
Authors
Banihashemi, Nahid
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thesis
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Abstract
The state- and control-constrained optimal control problems play an important role in science and engineering. Solving these problems is in general difficult because they are infinite-dimensional. In the first part of this thesis, we address this issue by solving state- and controlconstrained optimal control problems numerically. For doing this, we adapt an optimization method called Inexact Restoration. However, the computational effort needed for solving accurately these problems may be high. To alleviate this situation, we devise in the second part of this thesis a new algorithm which combines Inexact Restoration ideas with Adaptive Mesh Refinement.
In Chapter 1 we give a survey on numerical approaches for optimal control problems, as well as some preliminaries on optimal control theory. Chapters 2 and 3 describe the current background on the applications of Inexact Restoration. Namely, Chapter 2 reviews the use of Inexact Restoration for finitedimensional optimization, while Chapter 3 describes the application of this method for solving unconstrained optimal control problems. Chapters 4 and 5 include the novelties of this thesis. Chapter 4 extends the use of Inexact Restoration to state and control constrained optimal control problems. First, we convert the problem into a finite-dimensional optimization one by using the Euler discretization scheme. Second, we solve this finite dimensional problem using Inexact Restoration, and prove convergence of these solutions to a continuous-time solution of the original problem.
The Inexact Restoration algorithm we used to solve the discretized problem in Chapter 4 is designed over a fixed mesh. To improve the accuracy of the solutions obtained by this algorithm, the mesh needs to be made finer. However, a finer mesh incurs an increase in computational time and results in a larger-scale problem. This makes computations difficult, especially in the early iterations. In Chapter 5, we propose a new Inexact Restoration algorithm for solving the discretized problem which addressed these two issues. The problem we consider is still the same Euler discretized form used in Chapter 4. This new algorithm starts with a coarse mesh, which typically involves fewer discretization points than the fine mesh. The coarse mesh is then refined progressively by using the sufficient conditions of convergence of the Inexact Restoration method. This new algorithm is convergent to a fine mesh solution, by virtue of convergence of the Inexact Restoration method. We illustrate the algorithm on the container crane problem. Our experiments demonstrate significant computational savings and more robustness, when compared to the fixed-mesh algorithm of Chapter 4.
School/Discipline
School of Mathematics and Statistics
Dissertation Note
Thesis (PhDMathematics)--University of South Australia, 2012.
Provenance
Copyright 2012 Nahid Banihashemi. This work is made available under the Creative Commons Attribution-NonCommercial-NoDerivs Australia 3.0 licence (http://creativecommons.org/licenses/by-nc-nd/3.0/au/)
Description
xix, 160 pages
color illustrations
Includes bibliographical references (pages 149-160)
color illustrations
Includes bibliographical references (pages 149-160)
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