Optimal Control of ODEs and DAEs (eBook)

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Matthias Gerdts, Universität der Bundeswehr München, Germany.

Preface 6
Contents 8
1 Introduction 12
1.1 DAE Optimal Control Problems 19
1.1.1 Perturbation Index 35
1.1.2 Consistent Initial Values 41
1.1.3 Index Reduction and Stabilization 43
1.2 Transformation Techniques 47
1.2.1 Transformation to Fixed Time Interval 47
1.2.2 Transformation to Autonomous Problem 48
1.2.3 Transformation of Tschebyscheff Problems 49
1.2.4 Transformation of L 1 -Minimization Problems 49
1.2.5 Transformation of Interior-Point Constraints 50
1.3 Overview 53
1.4 Exercises 55
2 Infinite Optimization Problems 61
2.1 Function Spaces 61
2.1.1 Topological Spaces, Banach Spaces, and Hilbert Spaces 62
2.1.2 Mappings and Dual Spaces 65
2.1.3 Derivatives, Mean-Value Theorem, and Implicit Function Theorem 67
2.1.4 Lp-Spaces, Wq P-Spaces, Absolutely Continuous Functions, Functions of Bounded Variation
2.2 The DAE Optimal Control Problem as an Infinite Optimization Problem 79
2.3 Necessary Conditions for Infinite Optimization Problems 86
2.3.1 Existence of a Solution 88
2.3.2 Conic Approximation of Sets 90
2.3.3 Separation Theorems 95
2.3.4 First Order Necessary Optimality Conditions of Fritz John Type 98
2.3.5 Constraint Qualifications and Karush–Kuhn–Tucker Conditions 106
2.4 Exercises 111
3 Local Minimum Principles 115
3.1 Problems without Pure State and Mixed Control-State Constraints 117
3.1.1 Representation of Multipliers 122
3.1.2 Local Minimum Principle 125
3.1.3 Constraint Qualifications and Regularity 129
3.2 Problems with Pure State Constraints 136
3.2.1 Representation of Multipliers 138
3.2.2 Local Minimum Principle 141
3.2.3 Finding Controls on Active State Constraint Arcs 146
3.2.4 Jump Conditions for the Adjoint 148
3.3 Problems with Mixed Control-State Constraints 152
3.3.1 Representation of Multipliers 154
3.3.2 Local Minimum Principle 156
3.4 Summary of Local Minimum Principles for Index-One Problems 159
3.5 Exercises 163
4 Discretization Methods for ODEs and DAEs 168
4.1 Discretization by One-Step Methods 170
4.1.1 The Euler Method 170
4.1.2 Runge–Kutta Methods 173
4.1.3 General One-Step Method 179
4.1.4 Consistency, Stability, and Convergence of One-Step Methods 179
4.2 Backward Differentiation Formulas (BDF) 187
4.3 Linearized Implicit Runge–Kutta Methods 189
4.4 Automatic Step-size Selection 196
4.5 Computation of Consistent Initial Values 202
4.5.1 Projection Method for Consistent Initial Values 203
4.5.2 Consistent Initial Values via Relaxation 204
4.6 Shooting Techniques for Boundary Value Problems 206
4.6.1 Single Shooting Method using Projections 207
4.6.2 Single Shooting Method using Relaxations 214
4.6.3 Multiple Shooting Method 215
4.7 Exercises 219
5 Discretization of Optimal Control Problems 226
5.1 Direct Discretization Methods 227
5.1.1 Full Discretization Approach 229
5.1.2 Reduced Discretization Approach 231
5.1.3 Control Discretization 233
5.2 A Brief Introduction to Sequential Quadratic Programming 238
5.2.1 Lagrange–Newton Method 238
5.2.2 Sequential Quadratic Programming (SQP) 240
5.3 Calculation of Derivatives for Reduced Discretization 249
5.3.1 Sensitivity Equation Approach 250
5.3.2 Adjoint Equation Approach: The Discrete Case 252
5.3.3 Adjoint Equation Approach : The Continuous Case 259
5.4 Discrete Minimum Principle and Approximation of Adjoints 265
5.4.1 Example 273
5.5 An Overview on Convergence Results 284
5.5.1 Convergence of the Euler Discretization 284
5.5.2 Higher Order of Convergence for Runge–Kutta Discretizations 287
5.6 Numerical Examples 289
5.7 Exercises 299
6 Real-Time Control 303
6.1 Parametric Sensitivity Analysis and Open-Loop Real-Time Control 304
6.1.1 Parametric Sensitivity Analysis of Nonlinear Optimization Problems 304
6.1.2 Open-Loop Real-Time Control via Sensitivity Analysis 314
6.2 Feedback Controller Design by Optimal Control Techniques 326
6.3 Model Predictive Control 336
6.4 Exercises 343
7 Mixed-Integer Optimal Control 349
7.1 Global Minimum Principle 350
7.1.1 Singular Controls 362
7.2 Variable Time Transformation Method 370
7.3 Switching Costs, Dynamic Programming, Bellman’s Optimality Principle 384
7.3.1 Dynamic Optimization Model with Switching Costs 385
7.3.2 A Dynamic Programming Approach 387
7.3.3 Examples 394
7.4 Exercises 402
8 Function Space Methods 405
8.1 Gradient Method 406
8.2 Lagrange–Newton Method 422
8.2.1 Computation of the Search Direction 428
8.3 Exercises 440
Bibliography 444
Index 466