Optimization by variational methods

Cover......Page 1 Title......Page 2 Preface......Page 3 Contents......Page 7 OPTIMIZATION AND ENGINEERING PRACTICE......Page 13 BIBLIOGRAPHICAL NOTES......Page 14 1.2 The Simplest Problem......Page 16 1.3 A Variational Derivation......Page 19 1.4 An Optimal-control Problem: Formulation......Page 22 1.5 Optimal Proportional Control......Page 24 1.6 Discrete Optimal Proportional Control......Page 25 1.7 Discrete Optimal Control......Page 27 1.8 Lagrange Multipliers......Page 30 1.9 A Geometrical Example......Page 33 1.10 Discrete Proportional Control with Lagrange Multipliers......Page 35 1.11 Optimal Design of Multistage Systems......Page 36 1.12 Optimal Temperatures for Consecutive Reactions......Page 39 1.13 One-dimensional Processes......Page 41 1.14 An Inverse Problem......Page 42 1.15 Meaning of the Lagrange Multipliers......Page 44 1.16 Penalty Functions......Page 46 APPENDIX 1.1 Linear Difference Equations......Page 48 BIBLIOGRAPHICAL NOTES......Page 50 PROBLEMS......Page 52 2.1 Introduction......Page 56 2.2 Solution of Algebraic Equations......Page 57 2.3 An Application of the Newton-Raphson Method......Page 58 2.4 Fibonacci Search......Page 61 2.5 Steep Descent......Page 64 2.6 A Geometric Interpretation......Page 65 2.7 An Application of Steep Descent......Page 67 2.8 The Weighting Matrix......Page 69 2.9 Approximation to Steep Descent......Page 71 APPENDIX 2.1 Optimality of Fibonacci Search......Page 74 APPENDIX 2.2 Linear Programming......Page 77 BIBLIOGRAPHICAL NOTES......Page 80 PROBLEMS......Page 82 3.2 Euler Equation -......Page 85 3.3 Brachistochrone......Page 89 3.4 Optimal Linear Control......Page 91 3.11 A Distributed System......Page 103 3.6 Integral Constraints......Page 96 3.7 Maximum Area......Page 97 3.8 An Inverse Problem......Page 98 3.9 The Ritz-Galerkin Method......Page 100 3.10 An Eigenvalue Problem......Page 102 3.12 Control of a Distributed Plant......Page 105 BIBLIOGRAPHICAL NOTES......Page 108 PROBLEMS......Page 109 4.1 Introduction......Page 112 4.2 Variational Equations......Page 113 4.3 First Necessary Conditions......Page 117 4.4 Euler Equation......Page 120 4.5 Relation to Classical Mechanics......Page 121 4.6 Some Physical Equations and Useful Transformations......Page 122 4.7 Linear Feedback Control......Page 126 4.8 An Approximate Solution......Page 129 4.9 Control with Continuous Disturbances......Page 133 4.10 Proportional Plus Reset Control......Page 136 4.11 Optimal-yield Problems......Page 137 4.12 Optimal Temperatures for Consecutive Reactions......Page 140 BIBLIOGRAPHICAL NOTES......Page 142 PROBLEMS......Page 144 5.1 Introduction......Page 147 5.2 Necessary Conditions......Page 148 5.3 A Bang-bang Control Problem......Page 150 5.4 A Problem of Nonuniqueness......Page 154 5.5 Time-optimal Control of a Stirred-tank Reactor......Page 156 5.6 Nonlinear Time Optimal Control......Page 162 5.7 . Time-optimal Control of Underdamped Systems......Page 164 5.8 A Time-and-fuel Optimal Problem......Page 167 5.9 A Minimum-integral-square-error Criterion and Singular Solutions......Page 171 5.10 Nonlinear Minimum-integral-square-error Control......Page 175 5.11 Optimal Cooling Rate in Batch and Tubular Reactors......Page 177 BIBLIOGRAPHICAL NOTES......Page 181 PROBLEMS......Page 183 6.1 Introduction......Page 187 6.2 Integrating Factors and Green's Functions......Page 188 6.3 First-order Variational Equations......Page 192 6.4 The Minimization Problem and First Variation of the Objective......Page 193 6.5 The Weak Minimum Principle......Page 196 6.6 Equivalent Formulations......Page 199 6.7 An Application with Transversality Conditions......Page 201 6.8 The Strong Minimum Principle......Page 203 6.9 The Strong Minimum Principle: A Second Derivation......Page 206 6.10 Optimal Temperatures for Consecutive Reactions......Page 209 6.11 Optimality of the Steady State......Page 211 6.12 Optimal Operation of a Catalytic Reformer......Page 214 6.14 Necessary Condition for Singular Solutions......Page 219 6.15 Mixed Constraints......Page 221 6.16 State-variable Constraints......Page 223 6.17 Control with Inertia......Page 224 6.18 Discontinuous Multipliers......Page 226 6.19 Bottleneck Problems......Page 229 6.20 Sufficiency......Page 232 APPENDIX 6.1 Continuous Dependence of Solutions......Page 233 BIBLIOGRAPHICAL NOTES......Page 234 PROBLEMS......Page 238 7.1 Introduction......Page 240 7.2 Green's Functions......Page 241 7.3 The First Variation......Page 243 7.4 The Weak Minimum Principle......Page 244 7.5 Lagrange Multipliers......Page 245 7.6 Optimal Temperatures for Consecutive Reactions......Page 246 7.7 The Strong Minimum Principle: A Counterexample......Page 249 7.8 Second-order Variational Equations......Page 250 7.9 Mixed and State-variable Constraints......Page 254 BIBLIOGRAPHICAL NOTES......Page 255 PROBLEMS......Page 257 8.1 Introduction......Page 259 8.2 Linear Servomechanism Problem......Page 260 8.3 Three-mode Control......Page 262 8.4 Instantaneously Optimal Relay Control......Page 266 8.5 An Inverse Problem......Page 269 8.6 Discrete Linear Regulator......Page 274 APPENDIX 8.1 Liapunov Stability......Page 277 BIBLIOGRAPHICAL NOTES......Page 278 PROBLEMS......Page 281 9.2 Newton-Raphson Boundary Iteration......Page 283 9.3 Optimal Temperature Profile by Newton-Raphson Boundary Iteration......Page 286 9.4 Steep-descent Boundary Iteration......Page 290 9.5 Newton-Raphson Function Iteration: A Special Case......Page 295 9.6 Newton-Raphson Function Iteration: General Algorithm......Page 300 9.7 Optimal Pressure Profile by Newton-Raphson Function Iteration......Page 302 9.8 General Comments on Indirect Methods......Page 305 9.9 Steep Descent......Page 307 9.10 Steep Descent: Optimal Pressure Profile......Page 311 9.11 Steep Descent: Optimal Temperature Profile......Page 313 9.12 Steep Descent: Optimal Staged Temperatures......Page 316 9.13 Gradient Projection for Constrained End Points......Page 320 9.14 Min H......Page 323 9.15 Second-order Effects......Page 326 9.16 Second Variation......Page 327 BIBLIOGRAPHICAL NOTES......Page 333 PROBLEMS......Page 337 10.1 Introduction......Page 338 10.2 Recycle Processes......Page 339 10.3 Chemical Reaction with Recycle......Page 341 10.4 An Equivalent Formulation......Page 343 10.5 Lagrange Multipliers......Page 344 10.6 The General Analysis......Page 345 10.7 Reaction, Extraction, and Recycle......Page 349 10.8 Periodic Processes......Page 360 10.9 Decomposition......Page 367 BIBLIOGRAPHICAL NOTES......Page 369 PROBLEMS......Page 370 11.2 A Diffusion Process......Page 371 11.3 Variational Equations......Page 372 11.4 The Minimum Principle......Page 374 11.5 Linear Heat Conduction......Page 376 11.6 Steep Descent......Page 377 11.7 Computation for Linear Heat Conduction......Page 378 11.8 Chemical Reaction with Radial Diffusion......Page 383 11.9 Linear Feedforward-Feedback Control......Page 389 11.10 Optimal Feed Distribution in Parametric Pumping......Page 393 BIBLIOGRAPHICAL NOTES......Page 399 PROBLEMS......Page 401 12.1 I ntroduction......Page 404 12.2 The Principle of Optimality and Comnutation......Page 405 12.3 Optimal Temperature Sequences......Page 406 12.4 The Hamilton-Jacobi-Bellman Equation......Page 410 12.5 A Solution of the Hamilton-Jacobi-Bellman Equation......Page 412 12.6 The Continuous Hamilton-Jacobi-Bellman Equation......Page 414 12.7 The Linear Regulator Problem......Page 417 PROBLEMS......Page 419 Name Index......Page 423 Subject Index......Page 427