Variational calculus and optimal control optimization with elementary convexity
"This book supplies a broad-based introduction to variational methods for formulating and solving problems in mathematics and the applied sciences. It refines and extends the author's earlier text on variational calculus and a supplement on optimal control. It is the only current introduct...
Gespeichert in:
| Vorheriger Titel: | Troutman, John L. Variational calculus with elementary convexity |
|---|---|
| 1. Verfasser: | |
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
1996
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Undergraduate texts in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
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| 100 | 1 | |a Troutman, John L. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Variational calculus and optimal control |b optimization with elementary convexity |c John L. Troutman |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 1996 | |
| 300 | |a XV, 461 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Undergraduate texts in mathematics | |
| 520 | 1 | |a "This book supplies a broad-based introduction to variational methods for formulating and solving problems in mathematics and the applied sciences. It refines and extends the author's earlier text on variational calculus and a supplement on optimal control. It is the only current introductory text that uses elementary partial convexity of differentiable functions to characterize directly the solutions of some minimization problems before exploring necessary conditions for optimality or field theory methods of sufficiency. Through effective notation, it combines rudiments of analysis in (normed) linear spaces with simpler aspects of convexity to offer a multilevel strategy for handling such problems. It also employs convexity considerations to broaden the discussion of Hamilton's principle in mechanics and to introduce Pontjragin's principle in optimal control. It is mathematically self-contained but it uses applications from many disciplines to provide a wealth of examples and exercises. The book is accessible to upper-level undergraduates and should help its user understand theories of increasing importance in a society that values optimal performance."--BOOK JACKET. | |
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Datensatz im Suchindex
| DE-BY-TUM_call_number | 0202/MAT 490f 2002 A 938(2) 0001/96 A 1583 0102/MAT 490f 2001 A 29661(2)+2 0102/MAT 490f 2001 A 29661(2) |
|---|---|
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| DE-BY-TUM_media_number | 040020470008 040020659016 040001039492 040010581329 |
| _version_ | 1816711744098140160 |
| adam_text | Contents
Preface vii
CHAPTER 0
Review of Optimization in Ud 1
Problems 7
PART ONE
BASIC THEORY 11
CHAPTER 1
Standard Optimization Problems 13
1.1. Geodesic Problems 13
(a) Geodesies in M* 14
(b) Geodesies on a Sphere 15
(c) Other Geodesic Problems 17
1.2. Time of Transit Problems 17
(a) The Brachistochrone 17
(b) Steering and Control Problems 20
1.3. Isoperimetric Problems 21
1.4. Surface Area Problems 24
(a) Minimal Surface of Revolution 24
(b) Minimal Area Problem 25
(c) Plateau s Problem 26
1.5. Summary: Plan of the Text 26
Notation: Uses and Abuses 29
Problems 31
xi
xii Contents
CHAPTER 2
Linear Spaces and Gateaux Variations 36
2.1. Real Linear Spaces 36
2.2. Functions from Linear Spaces 38
2.3. Fundamentals of Optimization 39
Constraints 41
Rotating Fluid Column 42
2.4. The Gateaux Variations 45
Problems 50
CHAPTER 3
Minimization of Convex Functions 53
3.1. Convex Functions 54
3.2. Convex Integral Functions 56
Free End Point Problems 60
3.3. [Strongly] Convex Functions 61
3.4. Applications 65
(a) Geodesies on a Cylinder 65
(b) A Brachistochrone 66
(c) A Profile of Minimum Drag 69
(d) An Economics Problem 72
(e) Minimal Area Problem 74
3.5. Minimization with Convex Constraints 76
The Hanging Cable 78
Optimal Performance 81
3.6. Summary: Minimizing Procedures 83
Problems 84
CHAPTER 4
The Lemmas of Lagrange and Du Bois Reymond 97
Problems 101
CHAPTER 5
Local Extrema in Normed Linear Spaces 103
5.1. Norms for Linear Spaces 103
5.2. Normed Linear Spaces: Convergence and Compactness 106
5.3. Continuity 108
5.4. (Local) Extremal Points 114
5.5. Necessary Conditions: Admissible Directions 115
5.6*. Affine Approximation: The Frechet Derivative 120
Tangency 127
5.7. Extrema with Constraints: Lagrangian Multipliers 129
Problems 139
CHAPTER 6
The Euler Lagrange Equations 145
6.1. The First Equation: Stationary Functions 147
6.2. Special Cases of the First Equation 148
Contents xiii
(a) When/ = /(z) 149
(b) When/ = /(*, z) 149
(c) When f = f(y,z) 150
6.3. The Second Equation 153
6.4. Variable End Point Problems: Natural Boundary Conditions 156
Jakob Bernoulli s Brachistochrone 156
Transversal Conditions* 157
6.5. Integral Constraints: Lagrangian Multipliers 160
6.6. Integrals Involving Higher Derivatives 162
Buckling of a Column under Compressive Load 164
6.7. Vector Valued Stationary Functions 169
The Isoperimetric Problem 171
Lagrangian Constraints* 173
Geodesies on a Surface 177
6.8*. Invariance of Stationarity 178
6.9. Multidimensional Integrals 181
Minimal Area Problem 184
Natural Boundary Conditions 185
Problems 186
PART TWO
ADVANCED TOPICS 195
CHAPTER 7
Piecewise C1 Extremal Functions 197
7.1. Piecewise C1 Functions 198
(a) Smoothing 199
(b) Norms for C1 201
7.2. Integral Functions on C1 202
7.3. Extremals in C1 [a, by. The Weierstrass Erdmann Corner Conditions 204
A Sturm Liouville Problem 209
7.4. Minimization Through Convexity 211
Internal Constraints 212
7.5. Piecewise C1 Vector Valued Extremals 215
Minimal Surface of Revolution 217
Hilbert s Differentiability Criterion* 220
7.6*. Conditions Necessary for a Local Minimum 221
(a) The Weierstrass Condition 222
(b) The Legendre Condition 224
Bolza s Problem 225
Problems 227
CHAPTER 8
Variational Principles in Mechanics 234
8.1. The Action Integral 235
8.2. Hamilton s Principle: Generalized Coordinates 236
Bernoulli s Principle of Static Equilibrium 239
xiv Contents
8.3. The Total Energy 240
Spring Mass Pendulum System 241
8.4. The Canonical Equations 243
8.5. Integrals of Motion in Special Cases 247
Jacobi s Principle of Least Action 248
Symmetry and Invariance 250
8.6. Parametric Equations of Motion 250
8.7*. The Hamilton Jacobi Equation 251
8.8. Saddle Functions and Convexity; Complementary Inequalities 254
The Cycloid Is the Brachistochrone 257
Dido s Problem 258
8.9. Continuous Media 260
(a) Taut String 260
The Nonuniform String 264
(b) Stretched Membrane 266
Static Equilibrium of (Nonplanar) Membrane 269
Problems 270
CHAPTER 9*
Sufficient Conditions for a Minimum 282
9.1. The Weierstrass Method 283
9.2. [Strict] Convexity of f(x, Y, Z) 286
9.3. Fields 288
Exact Fields and the Hamilton Jacobi Equation* 293
9.4. Hilbert s Invariant Integral 294
The Brachistochrone* 296
Variable End Point Problems 297
9.5. Minimization with Constraints 300
The Wirtinger Inequality 304
9.6*. Central Fields 308
Smooth Minimal Surface of Revolution 312
9.7. Construction of Central Fields with Given Trajectory:
The Jacobi Condition 314
9.8. Sufficient Conditions for a Local Minimum 319
(a) Pointwise Results 320
Hamilton s Principle 320
(b) Trajectory Results 321
9.9*. Necessity of the Jacobi Condition 322
9.10. Concluding Remarks 327
Problems 329
PART THREE
OPTIMAL CONTROL 339
CHAPTER 10*
Control Problems and Sufficiency Considerations 341
10.1. Mathematical Formulation and Terminology 342
Contents xv
10.2. Sample Problems 344
(a) Some Easy Problems 345
(b) A Bolza Problem 347
(c) Optimal Time of Transit 348
(d) A Rocket Propulsion Problem 350
(e) A Resource Allocation Problem 352
(f) Excitation of an Oscillator 355
(g) Time Optimal Solution by Steepest Descent 357
10.3. Sufficient Conditions Through Convexity 359
Linear State Quadratic Performance Problem 361
10.4. Separate Convexity and the Minimum Principle 365
Problems 372
CHAPTER 11
Necessary Conditions for Optimality 378
11.1. Necessity of the Minimum Principle 378
(a) Effects of Control Variations 380
(b) Autonomous Fixed Interval Problems 384
Oscillator Energy Problem 389
(c) General Control Problems 391
11.2. Linear Time Optimal Problems 397
Problem Statement 398
A Free Space Docking Problem 401
11.3. General Lagrangian Constraints 404
(a) Control Sets Described by Lagrangian Inequalities 405
(b)* Variational Problems with Lagrangian Constraints 406
(c) Extensions 410
Problems 413
Appendix
A.0. Compact Sets in Rd 419
A.I. The Intermediate and Mean Value Theorems 421
A.2. The Fundamental Theorem of Calculus 423
A.3. Partial Integrals: Leibniz Formula 425
A.4. An Open Mapping Theorem 427
A. 5. Families of Solutions to a System of Differential Equations 429
A.6. The Rayleigh Ratio 435
A.7*. Linear Functionals and Tangent Cones in Rd 441
Bibliography 445
Historical References 450
Answers to Selected Problems 452
Index 457
|
| any_adam_object | 1 |
| author | Troutman, John L. |
| author_facet | Troutman, John L. |
| author_role | aut |
| author_sort | Troutman, John L. |
| author_variant | j l t jl jlt |
| building | Verbundindex |
| bvnumber | BV010617259 |
| callnumber-first | Q - Science |
| callnumber-label | QA315 |
| callnumber-raw | QA315.T724 1996 |
| callnumber-search | QA315.T724 1996 |
| callnumber-sort | QA 3315 T724 41996 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 660 SK 870 |
| classification_tum | MAT 525f MAT 490f |
| ctrlnum | (OCoLC)32236221 (DE-599)BVBBV010617259 |
| dewey-full | 515/.64 515/.6420 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.64 515/.64 20 |
| dewey-search | 515/.64 515/.64 20 |
| dewey-sort | 3515 264 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV010617259 |
| illustrated | Illustrated |
| indexdate | 2024-11-25T17:14:19Z |
| institution | BVB |
| isbn | 0387945113 9781461268871 |
| language | English |
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| open_access_boolean | |
| owner | DE-703 DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-898 DE-BY-UBR DE-355 DE-BY-UBR DE-384 DE-824 DE-29T DE-521 DE-634 DE-11 DE-188 DE-19 DE-BY-UBM |
| owner_facet | DE-703 DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-898 DE-BY-UBR DE-355 DE-BY-UBR DE-384 DE-824 DE-29T DE-521 DE-634 DE-11 DE-188 DE-19 DE-BY-UBM |
| physical | XV, 461 S. graph. Darst. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | Springer |
| record_format | marc |
| series2 | Undergraduate texts in mathematics |
| spellingShingle | Troutman, John L. Variational calculus and optimal control optimization with elementary convexity Convexe ruimten gtt Optimaliseren gtt Variatierekening gtt Calculus of variations Control theory Mathematical optimization Convex functions Konvexe Optimierung (DE-588)4137027-2 gnd Variationsrechnung (DE-588)4062355-5 gnd Konvexe Funktion (DE-588)4139679-0 gnd Optimale Kontrolle (DE-588)4121428-6 gnd |
| subject_GND | (DE-588)4137027-2 (DE-588)4062355-5 (DE-588)4139679-0 (DE-588)4121428-6 |
| title | Variational calculus and optimal control optimization with elementary convexity |
| title_auth | Variational calculus and optimal control optimization with elementary convexity |
| title_exact_search | Variational calculus and optimal control optimization with elementary convexity |
| title_full | Variational calculus and optimal control optimization with elementary convexity John L. Troutman |
| title_fullStr | Variational calculus and optimal control optimization with elementary convexity John L. Troutman |
| title_full_unstemmed | Variational calculus and optimal control optimization with elementary convexity John L. Troutman |
| title_old | Troutman, John L. Variational calculus with elementary convexity |
| title_short | Variational calculus and optimal control |
| title_sort | variational calculus and optimal control optimization with elementary convexity |
| title_sub | optimization with elementary convexity |
| topic | Convexe ruimten gtt Optimaliseren gtt Variatierekening gtt Calculus of variations Control theory Mathematical optimization Convex functions Konvexe Optimierung (DE-588)4137027-2 gnd Variationsrechnung (DE-588)4062355-5 gnd Konvexe Funktion (DE-588)4139679-0 gnd Optimale Kontrolle (DE-588)4121428-6 gnd |
| topic_facet | Convexe ruimten Optimaliseren Variatierekening Calculus of variations Control theory Mathematical optimization Convex functions Konvexe Optimierung Variationsrechnung Konvexe Funktion Optimale Kontrolle |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007083225&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT troutmanjohnl variationalcalculusandoptimalcontroloptimizationwithelementaryconvexity |