Numerical methods for optimal control

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1 Introduction 1
1 The Calculus of Variations 1
2 Optimal Control 5
3 Numerical Methods for Optimal Control Problems 7
2 Estimates on Solutions to Differential Equations and Their Approximations 13
1 Linear Approximations 13
2 Lagrangian, Hamiltonian and Reduced Gradients 19
3 First Order Method 27
1 Introduction 27
2 Representation of Functional Directional Derivatives .... 31
3 Relaxed Controls 32
4 The Algorithm 34
5 Convergence Properties of the Algorithm 38
6 Proof of the Convergence Theorem, etc 41
7 Concluding Remarks 52
4 Implementation 55
1 Implementable Algorithm 55
1.1 Second Order Correction To the Line Search .... 65
1.2 Resetting the Penalty Parameter 66
2 Semi-Infinite Programming Problem 66
3 Numerical Examples 68
5 Second Order Method 81
1 Introduction 81
2 Function Space Algorithm 84
3 Semi-Infinite Programming Method 86
4 Bounding the Number of Constraints 92
4.1 Some Remarks on Direction Finding Subproblems 94
4.2 The Nonlinear Programming Problem 98
XII
4.3 The Watchdog Technique for Redundant Constraints 107
4.4 Two-Step Superlinear Convergence 121
4.5 Numerical Experiments 125
5 Concluding Remarks 127
6 Runge—Kutta Based Procedure for Optimal Control of Differential — Algebraic Equations 129
1 Introduction 129
2 The Method 133
2.1 Implicit Runge-Kutta Methods 134
2.2 Calculation of the Reduced Gradients 137
3 Implementation of the Implicit Runge-Kutta Method . . . 144
3.1 Simplified Newton Iterations 144
3.2 Stopping Criterion for the Newton Method 145
3.3 Stepsize Selection 146
4 Numerical Experiments 151
5 Some Remarks on Integration and Optimization Accuracies 164
6 Concluding Remarks 166

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