Adaptive Numerical Solution of State Constrained Optimal Control Problems [Elektronische Ressource] / Olaf Benedix. Gutachter: Thomas Apel ; Boris Vexler. Betreuer: Boris Vexler

Sujets

• Mathematics

Informations

• Publié par: technische_universitat_munchen
• Publié le: 01 janvier 2011
• Nombre de lectures: 25
• Poids de l'ouvrage: 2 Mo

Extrait

Technische Universität München
Lehrstuhl für Mathematische Optimierung
Adaptive Numerical Solution of State Constrained
Optimal Control Problems
Olaf Benedix
Vollständiger Abdruck der von der Fakultät für Mathematik der Technischen Universität
München zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften (Dr.rer.nat.)
genehmigten Dissertation.
Vorsitzender: Univ.-Prof. Dr. Peter Rentrop
Prüfer der Dissertation: 1. Dr. Boris Vexler
2. Univ.-Prof. Dr. Thomas Apel
(Universität der Bundeswehr München)
Die Dissertation wurde am 14. 06. 2011 bei der Technischen Universität München eingereicht
und durch die Fakultät für Mathematik am 11. 07. 2011 angenommen.Contents
1. Introduction 1
2. Basic Concepts in Optimal Control 7
2.1. Problem setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2. State equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.3. State constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.4. Cost functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2. Existence and uniqueness of optimal solutions . . . . . . . . . . . . . . . . . . . 16
2.3. Discretization and optimization algorithms for problems without pointwise
constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.1. Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.2. Evaluation of derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.3. Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.4. Optimization methods for unconstrained problems . . . . . . . . . . . . 24
2.4. Treatment of inequality constraints . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5. A posteriori error estimation and adaptive algorithm . . . . . . . . . . . . . . . 30
3. Elliptic Optimal Control Problems with State Constraints 35
3.1. Analysis of the state equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2. Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3. Finite element discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.1. Discretization of the state variable . . . . . . . . . . . . . . . . . . . . . 41
3.3.2. of Lagrange multiplier and state constraint . . . . . . . . 44
3.3.3. Discretization of the control variable . . . . . . . . . . . . . . . . . . . . 45
3.3.4. Discrete optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4. Optimization with the primal-dual active set method . . . . . . . . . . . . . . . 47
3.5. A posteriori error estimator and adaptivity . . . . . . . . . . . . . . . . . . . . 52
3.6. Regularization and interior point method . . . . . . . . . . . . . . . . . . . . . 58
4. Parabolic Optimal Control Problems with State Constraints 61
4.1. Continous setting and optimality conditions . . . . . . . . . . . . . . . . . . . . 61
4.2. Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3. Finite element discretization in space and time . . . . . . . . . . . . . . . . . . 65
4.4. Optimization by interior point method . . . . . . . . . . . . . . . . . . . . . . . 68
4.5. A posteriori error estimator and adaptivity . . . . . . . . . . . . . . . . . . . . 70
5. Aspects of Implementation 79
iContents
5.1. Complete algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2. Implementation of Borel measures . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3. Possible modifications of the standard algorithm . . . . . . . . . . . . . . . . . 83
5.4. Considerations derived from practical problems . . . . . . . . . . . . . . . . . . 85
6. Numerical Results 89
6.1. Elliptic problem with known exact solution . . . . . . . . . . . . . . . . . . . . 90
6.2. with unknown exact . . . . . . . . . . . . . . . . . . . 94
6.3. Nonlinear elliptic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.4. Parabolic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7. Optimal Control of Young Concrete Thermo-Mechanical Properties 103
7.1. Problem introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2. Modelling the involved quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.3. State equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.4. Optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.4.1. State constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.4.2. Cost functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.5. Examples and numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.5.1. Control of initial temperature and heat transfer . . . . . . . . . . . . . . 118
7.5.2. Control of the concrete recipe . . . . . . . . . . . . . . . . . . . . . . . . 122
7.5.3. Control of the flow rate of a water cooling system . . . . . . . . . . . . . 124
8. Summary 129
Acknowledgments 131
A. Convergence order for the Laplace equation with irregular data 133
B. Utilized data for the models of the material properties of concrete 143
List of Figures 145
List of Tables 147
List of Algorithms 149
Bibliography 151
ii1. Introduction
The central subject of interest of this thesis is the class of optimal control problems with
partial differential equations and additional state constraints. The focus lies especially on
the construction of numerical solution algorithms to find an approximate solution to such a
problem, and the effectiveness of such algorithms.
The central problem class features many different ingredients, over which we give a short
overview here. The general problem form considered is

 minJ(q,u) q∈Q,u∈X
(P ) u =S(q) (1.1)
 G(u)≥ 0 .
Here,u is called the state function, searched for in the state spaceX, andq the control variable,
searched for in the control space Q. In the field of optimal control, X is usually considered
na function space. The domain of the state functions might be a spatial domain Ω⊂ R
(n∈{2, 3}) or in the case of time-dependent problems a domain in time and spaceI×Ω with a
given time interval I = (0,T ). The operator S is called control-to-state operator. It represents
the solution operator of a partial differential equation, which in turn is called the state equation.
In this thesis, elliptic and parabolic state equations are considered. The problem (P ) is then
called elliptic, or parabolic optimal control problem (OCP), respectively. The functional
J : Q×X→R is called the cost functional, and the function G is the constraint function for
the state. With all these ingredients present, (P ) is called a state constrained optimal control
problem. Without the conditionG(u)≥ 0 one would speak of an unconstrained control
problem, which can be regarded as the basis class of optimal control problems.
Unconstrained optimal control problems have been of interest in applied mathematics for some
time now. A lot of practical problems, their origin ranging from civil engineering via optics to
chemical engineering and biological applications, can be modeled as optimal control problems
with partial differential equations. This is not surprising, since most technical processes allow
for user input after the initial setup, and guiding the system’s output to a user-determined
configuration is a natural desire as well. Also understandable is the possible need for bounds
on input and output variables. For most technical problems, only certain amounts of input are
possible, and concerning the output, certain states might lead to catastrophic scenarios that
must be avoided at all cost.
This thesis deals with state constrained problems, which can be motivated in different ways.
From the viewpoint of the field of optimization, (P) can be seen as an optimization problem
on Q×X, with a partial differential equation as an equality constraint, and a pointwise
inequality constraint. The motivation to consider this problem class becomes possibly clearer
from an applicational point of view. Suppose that a scientific or technical process of interest
11. Introduction
is described by a partial differential equation. For notational purposes the quantities which
are considered influencable are gathered in the control variable q. On the other hand, the
quantities that are regarded as descriptive of the process’ status, are gathered in the state
function u. We think of the partial differential equation in such a way that u is the solution
depending on q, and thus write formally u =S(q). The quest is now to find the pair (q,u)
of a control q and corresponding state u =S(q) that is the most favorable to the user. By
means of the condition G(u)≥ 0 with a properly modeled function G the user can rule out
some pairs completely. Amongst the remaining pairs, favorability is determined by a given
functional J(q,u). This functional is modeled in such a way that a more favourable pair (q,u)
is mapped to a smaller value of J.
Optima