Calculus of variations

195 Pages·2011·1.164 MB·English

by  Miersemann E.

Calculus of Variations
Lecture Notes
Erich Miersemann
Department of Mathematics
Leipzig University
Version October 14, 2011
2
Contents
1 Introduction 9
1.1 Problems in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.1 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.2 Nash equilibrium . . . . . . . . . . . . . . . . . . . . . 10
1.1.3 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Ordinary differential equations . . . . . . . . . . . . . . . . . 11
1.2.1 Rotationally symmetric minimal surface . . . . . . . . 12
1.2.2 Brachistochrone . . . . . . . . . . . . . . . . . . . . . 13
1.2.3 Geodesic curves . . . . . . . . . . . . . . . . . . . . . . 14
1.2.4 Critical load . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.5 Euler’s polygonal method . . . . . . . . . . . . . . . . 20
1.2.6 Optimal control . . . . . . . . . . . . . . . . . . . . . . 21
1.3 Partial differential equations . . . . . . . . . . . . . . . . . . . 22
1.3.1 Dirichlet integral . . . . . . . . . . . . . . . . . . . . . 22
1.3.2 Minimal surface equation . . . . . . . . . . . . . . . . 23
1.3.3 Capillary equation . . . . . . . . . . . . . . . . . . . . 26
1.3.4 Liquid layers . . . . . . . . . . . . . . . . . . . . . . . 29
1.3.5 Extremal property of an eigenvalue . . . . . . . . . . . 30
1.3.6 Isoperimetric problems . . . . . . . . . . . . . . . . . . 31
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 Functions of n variables 39
2.1 Optima, tangent cones . . . . . . . . . . . . . . . . . . . . . . 39
2.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2 Necessary conditions . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.1 Equality constraints . . . . . . . . . . . . . . . . . . . 50
2.2.2 Inequality constraints . . . . . . . . . . . . . . . . . . 52
2.2.3 Supplement . . . . . . . . . . . . . . . . . . . . . . . . 56
2.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 58
3
4 CONTENTS
2.3 Sufficient conditions . . . . . . . . . . . . . . . . . . . . . . . 59
2.3.1 Equality constraints . . . . . . . . . . . . . . . . . . . 61
2.3.2 Inequality constraints . . . . . . . . . . . . . . . . . . 62
2.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4 Kuhn-Tucker theory . . . . . . . . . . . . . . . . . . . . . . . 65
2.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.5.1 Maximizing of utility . . . . . . . . . . . . . . . . . . . 72
2.5.2 V is a polyhedron . . . . . . . . . . . . . . . . . . . . 73
2.5.3 Eigenvalue equations . . . . . . . . . . . . . . . . . . . 73
2.5.4 Unilateral eigenvalue problems . . . . . . . . . . . . . 77
2.5.5 Noncooperative games . . . . . . . . . . . . . . . . . . 79
2.5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.6 Appendix: Convex sets . . . . . . . . . . . . . . . . . . . . . . 90
2.6.1 Separation of convex sets . . . . . . . . . . . . . . . . 90
2.6.2 Linear inequalities . . . . . . . . . . . . . . . . . . . . 94
2.6.3 Projection on convex sets . . . . . . . . . . . . . . . . 96
2.6.4 Lagrange multiplier rules . . . . . . . . . . . . . . . . 98
2.6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 101
2.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3 Ordinary differential equations 105
3.1 Optima, tangent cones, derivatives . . . . . . . . . . . . . . . 105
3.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.2 Necessary conditions . . . . . . . . . . . . . . . . . . . . . . . 109
3.2.1 Free problems . . . . . . . . . . . . . . . . . . . . . . . 109
3.2.2 Systems of equations . . . . . . . . . . . . . . . . . . . 120
3.2.3 Free boundary conditions . . . . . . . . . . . . . . . . 123
3.2.4 Transversality conditions . . . . . . . . . . . . . . . . 125
3.2.5 Nonsmooth solutions . . . . . . . . . . . . . . . . . . . 129
3.2.6 Equality constraints; functionals . . . . . . . . . . . . 134
3.2.7 Equality constraints; functions . . . . . . . . . . . . . 137
3.2.8 Unilateral constraints . . . . . . . . . . . . . . . . . . 141
3.2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 145
3.3 Sufficient conditions; weak minimizers . . . . . . . . . . . . . 149
3.3.1 Free problems . . . . . . . . . . . . . . . . . . . . . . . 149
3.3.2 Equality constraints . . . . . . . . . . . . . . . . . . . 152
3.3.3 Unilateral constraints . . . . . . . . . . . . . . . . . . 155
3.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 162
3.4 Sufficient condition; strong minimizers . . . . . . . . . . . . . 164
CONTENTS 5
3.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 170
3.5 Optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . 171
3.5.1 Pontryagin’s maximum principle . . . . . . . . . . . . 172
3.5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 172
3.5.3 Proof of Pontryagin’s maximum principle; free endpoint176
3.5.4 Proof of Pontryagin’s maximum principle; fixed end-
point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
3.5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 185
6 CONTENTS
Preface
These lecture notes are intented as a straightforward introduction to the
calculus of variations which can serve as a textbook for undergraduate and
beginning graduate students.
The main body of Chapter 2 consists of well known results concernig
necessary or sufficient criteria for local minimizers, including Lagrange mul-
tiplier rules, of real functions defined on a Euclidian n-space. Chapter 3
concerns problems governed by ordinary differential equations.
The content of these notes is not encyclopedic at all. For additional
reading we recommend following books: Luenberger [36], Rockafellar [50]
and Rockafellar and Wets [49] for Chapter 2 and Bolza [6], Courant and
Hilbert[9],GiaquintaandHildebrandt[19],JostandLi-Jost[26],Sagan[52],
Troutman [59] and Zeidler [60] for Chapter 3. Concerning variational prob-
lems governed by partial differential equations see Jost and Li-Jost [26] and
Struwe [57], for example.
7
8 CONTENTS
Chapter 1
Introduction
A huge amount of problems in the calculus of variations have their origin
in physics where one has to minimize the energy associated to the problem
under consideration. Nowadays many problems come from economics. Here
is the main point that the resources are restricted. There is no economy
without restricted resources.
Some basic problems in the calculus of variations are:
(i) find minimizers,
(ii) necessary conditions which have to satisfy minimizers,
(iii) find solutions (extremals) which satisfy the necessary condition,
(iv)sufficientconditionswhichguaranteethatsuchsolutionsareminimizers,
(v) qualitative properties of minimizers, like regularity properties,
(vi) how depend minimizers on parameters?,
(vii) stability of extremals depending on parameters.
In the following we consider some examples.
1.1 Problems in Rn
1.1.1 Calculus
Let f : V R, where V Rn is a nonempty set. Consider the problem
7→ ⊂
x V : f(x) f(y) for all y V.
∈ ≤ ∈
If there exists a solution then it follows further characterizations of the
solution which allow in many cases to calculate this solution. The main tool
9
10 CHAPTER 1. INTRODUCTION
for obtaining further properties is to insert for y admissible variations of x.
As an example let V be a convex set. Then for given y V
∈
f(x) f(x+²(y x))
≤ −
for all real 0 ² 1. From this inequality one derives the inequality
≤ ≤
f(x),y x 0 for all y V,
h∇ − i ≥ ∈
provided that f C1(Rn).
∈
1.1.2 Nash equilibrium
Ingeneralizationtotheoboveproblemweconsidertworealfunctionsf (x,y),
i
i = 1,2, defined on S S , where S Rmi. An (x∗,y∗) S S is called
1 2 i 1 2
× ⊂ ∈ ×
a Nash equlibrium if
f (x,y∗) f (x∗,y∗) for all x S
1 1 1
≤ ∈
f (x∗,y) f (x∗,y∗) for all y S .
2 2 2
≤ ∈
The functions f , f are called payoff functions of two players and the sets
1 2
S and S are the strategy sets of the players. Under additional assumptions
1 2
on f and S there exists a Nash equilibrium, see Nash [46]. In Section 2.4.5
i i
we consider more general problems of noncooperative games which play an
important role in economics, for example.
1.1.3 Eigenvalues
Consider the eigenvalue problem
Ax = λBx,
whereAandBarerealandsymmetricmatriceswithnrows(andncolumns).
Suppose that By,y > 0 for all y Rn 0 , then the lowest eigenvalue λ
1
h i ∈ \{ }
is given by
Ay,y
λ = min h i.
1
y∈Rn\{0} By,y
h i
The higher eigenvalues can be characterized by the maximum-minimum
principle of Courant, see Section 2.5.
In generalization, let C Rn be a nonempty closed convex cone with vertex
⊂
at the origin. Assume C = 0 . Then, see [37],
6 { }
Ay,y
λ = min h i
1
y∈C\{0} By,y
h i