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Calculus of Variations Calculus of Variations Lecture Notes Erich Miersemann Department of Mathematics Leipzig University Version October, 2012 2 Contents 1 Introduction 9 1.1 Problems in Rn .......................... 9 1.1.1 Calculus.......................... 9 1.1.2 Nashequilibrium . 10 1.1.3 Eigenvalues ........................ 10 1.2 Ordinarydifferentialequations . 11 1.2.1 Rotationally symmetric minimal surface . 12 1.2.2 Brachistochrone . 13 1.2.3 Geodesiccurves. 14 1.2.4 Criticalload ....................... 15 1.2.5 Euler’spolygonalmethod . 20 1.2.6 Optimalcontrol. 21 1.3 Partialdifferentialequations. 22 1.3.1 Dirichletintegral . 22 1.3.2 Minimal surface equation . 23 1.3.3 Capillaryequation . 26 1.3.4 Liquidlayers ....................... 29 1.3.5 Extremal property of an eigenvalue . 30 1.3.6 Isoperimetricproblems. 31 1.4 Exercises ............................. 33 2 Functions of n variables 39 2.1 Optima,tangentcones . 39 2.1.1 Exercises ......................... 45 2.2 Necessaryconditions . 47 2.2.1 Equalityconstraints . 49 2.2.2 Inequalityconstraints . 52 2.2.3 Supplement ........................ 56 2.2.4 Exercises ......................... 58 3 4 CONTENTS 2.3 Sufficientconditions . 59 2.3.1 Equalityconstraints . 61 2.3.2 Inequalityconstraints . 62 2.3.3 Exercises ......................... 64 2.4 Kuhn-Tuckertheory . 65 2.4.1 Exercises ......................... 71 2.5 Examples ............................. 72 2.5.1 Maximizingofutility. 72 2.5.2 Visapolyhedron .................... 73 2.5.3 Eigenvalueequations. 73 2.5.4 Unilateral eigenvalue problems . 77 2.5.5 Noncooperativegames . 79 2.5.6 Exercises ......................... 83 2.6 Appendix:Convexsets. 90 2.6.1 Separation of convex sets . 90 2.6.2 Linearinequalities . 94 2.6.3 Projectiononconvexsets . 96 2.6.4 Lagrangemultiplierrules . 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 Necessaryconditions . 109 3.2.1 Freeproblems. 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 Freeproblems. 149 3.3.2 Equalityconstraints . 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 .........................169 3.5 Optimalcontrol. 170 3.5.1 Pontryagin’s maximum principle . 171 3.5.2 Examples .........................171 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 concerning necessary or sufficient criteria for local minimizers, including Lagrange mul- tiplier rules, of real functions defined on a Euclidean 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], Giaquinta and Hildebrandt [19], Jost and Li-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) sufficient conditions which guarantee that such solutions are minimizers, (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 In generalization to the above problem we consider two real functions fi(x, y), i = 1, 2, defined on S S , where S Rmi . An (x∗, y∗) S S is called 1 × 2 i ⊂ ∈ 1 × 2 a Nash equilibrium 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 f1, f2 are called payoff functions of two players and the sets S1 and S2 are the strategy sets of the players. Under additional assumptions on fi and Si there exists a Nash equilibrium, see Nash [46]. In Section 2.4.5 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, where A and B are real and symmetric matrices with n rows (and n columns). Suppose that By,y > 0 for all y Rn 0 , then the lowest eigenvalue λ h i ∈ \ { } 1 is given by Ay, y λ1 = min h i . 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 λ1 = min h i y∈C\{0} By,y h i 1.2. ORDINARY DIFFERENTIAL EQUATIONS 11 is the lowest eigenvalue of the variational inequality x C : Ax, y x λ Bx,y x for all y C. ∈ h − i ≥ h − i ∈ Remark. A set C Rn is said to be a cone with vertex at x if for any ⊂ y C it follows that x + t(y x) C for all t> 0. ∈ − ∈ 1.2 Ordinary differential equations Set b E(v)= f(x,v(x),v0(x)) dx Za and for given u , u R a b ∈ V = v C1[a, b]: v(a)= u , v(b)= u , { ∈ a b} where <a<b< and f is sufficiently regular. One of the basic −∞ ∞ problems in the calculus of variation is (P ) minv∈V E(v). That is, we seek a u V : E(u) E(v) for all v V. ∈ ≤ ∈ Euler equation. Let u V be a solution of (P) and assume additionally ∈ u C2(a, b), then ∈ d 0 0 f 0 (x,u(x),u (x)) = f (x,u(x),u (x)) dx u u in (a, b). Proof. Exercise. Hints: For fixed φ C2[a, b] with φ(a) = φ(b) = 0 and ∈ real ², ² < ² , set g(²)= E(u + ²φ). Since g(0) g(²) it follows g0(0) = 0. | | 0 ≤ Integration by parts in the formula for g0(0) and the following basic lemma in the calculus of variations imply Euler’s equation. 2 12 CHAPTER 1. INTRODUCTION y u b u a a b x Figure 1.1: Admissible variations Basic lemma in the calculus of variations. Let h C(a, b) and ∈ b h(x)φ(x) dx = 0 Za for all φ C1(a, b). Then h(x) 0 on (a, b). ∈ 0 ≡ Proof. Assume h(x ) > 0 for an x (a, b), then there is a δ > 0 such that 0 0 ∈ (x δ, x + δ) (a, b) and h(x) h(x )/2 on (x δ, x + δ). Set 0 − 0 ⊂ ≥ 0 0 − 0 δ2 x x 2 2 if x (x δ, x + δ) φ(x)= −| − 0| ∈ 0 − 0 . 0 if x (a, b) [x δ, x + δ] ½ ¡ ¢ ∈ \ 0 − 0 Thus φ C1(a, b) and ∈ 0 b h(x ) x0+δ h(x)φ(x) dx 0 φ(x) dx > 0, ≥ 2 Za Zx0−δ which is a contradiction to the assumption of the lemma. 2 1.2.1 Rotationally symmetric minimal surface Consider a curve defined by v(x), 0 x l, which satisfies v(x) > 0 on [0,l] ≤ ≤ and v(0) = a, v(l) = b for given positive a and b, see Figure 1.2. Let (v) S 1.2. ORDINARY DIFFERENTIAL EQUATIONS 13 b S a l x Figure 1.2: Rotationally symmetric surface be the surface defined by rotating the curve around the x-axis. The area of this surface is l (v) = 2π v(x) 1 + (v0(x))2 dx. |S | 0 Z p Set V = v C1[0,l]: v(0) = a, v(l)= b, v(x) > 0 on (a, b) . { ∈ } Then the variational problem which we have to consider is min (v) . v∈V |S | Solutions of the associated Euler equation are catenoids (= chain curves), see an exercise.
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