Optimal Control for Chemical Engineers

Optimal Control for Chemical Engineers......Page 4
Contents......Page 8
Preface......Page 14
Function Vectors......Page 16
Derivatives Involving Vectors......Page 17
Miscellaneous Symbols......Page 18
1.1 Definition......Page 20
1.2 Optimal Control versus Optimization......Page 23
1.3.1 Batch Distillation......Page 24
1.3.2 Plug Flow Reactor......Page 25
1.3.3 Heat Exchanger......Page 27
1.3.4 Gas Diffusion in a Non-Volatile Liquid......Page 28
1.3.5 Periodic Reactor......Page 30
1.3.6 Nuclear Reactor......Page 31
1.3.7 Vapor Extraction of Heavy Oil......Page 32
1.3.8 Chemotherapy......Page 34
1.3.9 Medicinal Drug Delivery......Page 35
1.3.10 Blood Flow and Metabolism......Page 36
1.4 Structure of Optimal Control Problems......Page 38
Bibliography......Page 39
Exercises......Page 40
2.1 From Function to Functional......Page 42
2.2 Domain of a Functional......Page 44
2.2.2 Norm of a Function......Page 45
2.3 Properties of Functionals......Page 46
2.4.1 Fréchet Differential......Page 47
2.4.2 Gâteaux Differential......Page 49
2.4.3.1 Homogeneity of Variation......Page 55
2.4.5 Relations between Differentials......Page 57
2.5 Variation of an Integral Objective Functional......Page 59
2.5.1 Equivalence to Other Differentials......Page 60
2.5.1.1 Equivalence to the Fréchet Differential......Page 61
2.5.1.2 Equivalence to the Gâteaux Differential......Page 62
2.5.2 Application to Optimal Control Problems......Page 63
2.6.1 Second Degree Homogeneity......Page 69
2.6.2 Contribution to Functional Change......Page 70
2.A Second-Order Taylor Expansion......Page 71
Exercises......Page 73
3.1 Necessary Condition for Optimality......Page 76
3.2 Application to Simplest Optimal Control Problem......Page 77
3.2.2 Optimal Control Analysis......Page 78
3.2.3 Generalization......Page 86
3.3 Solving an Optimal Control Problem......Page 90
3.3.1 Presence of Several Local Optima......Page 92
3.4 Sufficient Conditions......Page 93
3.4.1 Weak Sufficient Condition......Page 94
3.5 Piecewise Continuous Controls......Page 95
3.A Differentiability of λ......Page 97
3.C Mangasarian Sufficiency Condition......Page 100
Bibliography......Page 102
Exercises......Page 103
4.1 Motivation......Page 106
4.3 Lagrange Multiplier Theorem......Page 107
4.3.1 Generalization to Several Equality Constraints......Page 113
4.3.2 Generalization to Several Functions......Page 115
4.3.2.1 Simplification of Constraint Qualification......Page 116
4.3.3 Application to Optimal Control Problems......Page 118
4.3.3.1 Serial Application of the Lagrange Multiplier Rule......Page 119
4.3.3.2 Generalization to Several States and Controls......Page 122
4.3.3.3 Presence of Algebraic Constraints......Page 124
4.4 Lagrange Multiplier and Objective Functional......Page 126
4.5 John Multiplier Theorem for Inequality Constraints......Page 128
4.5.1 Generalized John Multiplier Theorem......Page 132
4.A Inverse Function Theorem......Page 134
Bibliography......Page 139
Exercises......Page 140
5.1 Application......Page 142
5.2.1 Class of Controls......Page 145
5.2.3 Notation......Page 146
5.3.1 Assumptions......Page 147
5.4 Derivation of Pontryagin’s Minimum Principle......Page 148
5.4.1 Pulse Perturbation of Optimal Control......Page 150
5.4.3 Effect on Final State......Page 153
5.4.4 Choice of Final Costate......Page 155
5.4.5 Minimality of the Hamiltonian......Page 156
5.4.5.1 Minimality Even When û(t) Is Discontinuous......Page 157
5.A Convexity of Final States......Page 164
5.B Supporting Hyperplane of a Convex Set......Page 168
Bibliography......Page 171
6.1.1 Free Final State......Page 172
6.1.2 Fixed Final State......Page 176
6.1.3 Final State on Hypersurfaces......Page 177
6.2.2 Fixed Final State......Page 180
6.2.3 Final State on Hypersurfaces......Page 181
6.3.1 Algebraic Equality Constraints......Page 182
6.3.2 Algebraic Inequality Constraints......Page 185
6.4.1 Integral Equality Constraints......Page 187
6.4.2 Integral Inequality Constraints......Page 190
6.5 Interior Point Constraints......Page 191
6.6 Discontinuous Controls......Page 196
6.7 Multiple Integral Problems......Page 197
Bibliography......Page 201
Exercises......Page 202
7.1.1 Free Final Time and Free Final State......Page 204
7.1.2 Iterative Procedure......Page 205
7.1.3.1 Numerical Implementation......Page 206
7.1.4 Algorithm for the Gradient Method......Page 210
7.1.5 Fixed Final Time and Free Final State......Page 218
7.2.1 Free Final Time and Final State on Hypersurfaces......Page 220
7.2.2 Free Final Time but Fixed Final State......Page 226
7.2.3 Algebraic Equality Constraints......Page 228
7.2.4 Integral Equality Constraints......Page 233
7.2.5 Algebraic Inequality Constraints......Page 236
7.2.6 Integral Inequality Constraints......Page 240
7.3 Shooting Newton–Raphson Method......Page 242
7.A.1 Objective......Page 248
7.A.2 Sufficiency Check......Page 249
Exercises......Page 250
8.1 Optimality of Periodic Controls......Page 254
8.1.1 Necessary Conditions......Page 255
8.2.2 Shooting Newton–Raphson Method......Page 258
8.3 Pi Criterion......Page 267
8.4 Pi Criterion with Control Constraints......Page 275
8.A Necessary Conditions for Optimal Steady State......Page 279
8.B Derivation of Equation (8.12)......Page 280
8.C Fourier Transform......Page 283
Bibliography......Page 284
Exercises......Page 285
9.2 Continuity of a Function......Page 286
9.3 Intervals and Neighborhoods......Page 287
9.5.1 Big-O Notation......Page 288
9.7.1 Non-Autonomous to Autonomous Transformation......Page 289
9.8 Differential......Page 290
9.9.1 Directional Derivative......Page 291
9.11 Newton–Raphson Method......Page 292
9.13 Fundamental Theorem of Calculus......Page 294
9.15 Intermediate Value Theorem......Page 295
9.17 Bolzano–Weierstrass Theorem......Page 296
9.19 Linear or Vector Space......Page 297
9.20 Direction of a Vector......Page 298
9.22 Triangle Inequality for Integrals......Page 299
9.24 Operator Inequality......Page 300
9.25 Conditional Statement......Page 301
Bibliography......Page 302
Index......Page 304