Preface......Page 6
Contents......Page 8
About the Editors......Page 10
1 Introduction......Page 12
2.1 Problem Formulation......Page 15
2.2 Regularized Problem......Page 17
3 Dynamic Programming and Hamilton-Jacobi-Bellman Equation......Page 19
3.1 HJB Equation......Page 21
4 Numerical Approximation......Page 23
5.1 Example 1......Page 27
5.2 Example 2......Page 30
6 Conclusions......Page 31
References......Page 32
1 Introduction......Page 34
2 Problem Formulation and Solution Approach......Page 35
3 Upper Level: Route Planning......Page 37
4 Lower Level: Minimum Time Driving......Page 39
5 Levels Coupling......Page 42
6 Numerical Results......Page 43
6.1 General Evaluation of the Bi-level Algorithm......Page 44
6.2 Influence of the Parameters ΞΈ and Ξ»......Page 46
7 Conclusions......Page 47
References......Page 48
1 Introduction......Page 50
2 Optimal Control Problems with Time Delays and Optimality Conditions......Page 51
3 Discretization......Page 56
4 Convergence......Page 58
5 Numerical Simulations......Page 61
References......Page 73
1 Introduction......Page 74
2.1 The State Equation......Page 75
2.2 The State-Constrained Optimization Problem......Page 78
2.3 First-Order Optimality Conditions......Page 80
3 The Primal-Dual Active Set Strategy (PDASS)......Page 82
4 Proper Orthogonal Decomposition......Page 85
5 A-Posteriori Error Analysis......Page 86
6 Numerical Tests......Page 88
6.1 Test 1: Economic Optimal Control......Page 89
6.2 Test 2: Cost of Tracking Type......Page 92
7 Conclusions......Page 94
References......Page 96
Order Reduction Approaches for the Algebraic Riccati Equation and the LQR Problem......Page 99
1 Introduction......Page 100
2 The Linear-Quadratic Regulator Problem and Model Order Reduction......Page 102
3.1 Proper Orthogonal Decomposition......Page 104
3.2 Balanced Truncation......Page 105
4 Adaptive Reduction of the Algebraic Riccati Equation......Page 106
4.1 Galerkin and Petrov-Galerkin Riccati......Page 107
5 Numerical Experiments......Page 110
5.1 Test 1: 2D Linear Heat Equation......Page 111
5.2 Test 2: 2D Linear Convection-Diffusion Equation......Page 113
5.3 Test 3: A Discussion on Large Scale Balanced Truncation......Page 115
References......Page 117
1 Introduction......Page 120
2 Existence of Solution with Box and Sparse Constraints......Page 122
3.1 Finite Differences Discretization......Page 124
3.2.1 Box-Constrained Optimization Problems......Page 125
3.3 Fast and Reliable Solution of Sequences of FDEs......Page 127
3.3.1 Preconditioners for the Sequence of Jacobian in Semismooth Newton Method......Page 130
4 Numerical Examples......Page 135
4.1 Box-Constrained Problem......Page 136
4.2 Sparse Constrained Problem......Page 138
References......Page 143
Control, Shape, and Topological Derivatives via Minimax Differentiability of Lagrangians......Page 145
1 Introduction......Page 146
2 Examples of Derivatives with Respect to a Control, Shape, or Topological Variable......Page 147
2.2 Shape Derivative via the Velocity Method as a Differential......Page 148
2.3 Topological Derivative via Dilatations as a Semidifferential......Page 149
2.3.1 Tangent Space to the Group of Characteristic Functions......Page 150
2.3.2 The d-Dimensional Minkowski Content and d-Rectifiable Sets......Page 152
2.3.3 Back to the Example......Page 154
3.1 Abstract Framework......Page 155
3.2 Original Condition of Sturm and Its First Extension......Page 157
3.3 A New Condition with the Standard Adjoint at t=0......Page 159
3.4.1 Directional Derivative with Respect to the Control......Page 160
3.4.2 Shape Derivative......Page 162
3.4.3 Topological Derivative......Page 165
4 Minimax Theorems in the Mutivalued Case......Page 168
References......Page 172
1 Introduction......Page 173
2 Minimum Energy Estimation in Discrete Time......Page 175
3 Taylor Polynomial Approach......Page 178
4 The Relation Between the MEE and the EKF......Page 182
5 Example: Lorenz Attractor......Page 185
6 Conclusion......Page 188
References......Page 190
Probabilistic Max-Plus Schemes for Solving Hamilton-Jacobi-Bellman Equations......Page 191
1 Introduction......Page 192
2 The Probabilistic Time Discretization of Fahim, Touzi and Warin......Page 195
3 Monotone Probabilistic Approximation of First and Second Order Derivatives and Their Estimates......Page 196
4 Monotone Probabilistic Schemes for HJB Equations......Page 201
5 The Probabilistic Max-Plus Method......Page 209
References......Page 217
1 Introduction......Page 218
2 Optimal Control Problem......Page 220
3 Max-Plus Fundamental Solution Semigroups......Page 221
4 Adaptive Max-Plus Eigenvector Method......Page 229
5 Examples......Page 237
Appendix......Page 242
References......Page 246
1 Introduction......Page 248
2 Dequantization......Page 250
3.1 Stationarity Definitions......Page 251
3.2 The Non-inertial Frame......Page 252
3.3 Extensions to the Complex Domain......Page 253
4 An Expansion......Page 255
4.1 An Alternate Assumption......Page 258
5 Periodic S 0 Solutions......Page 260
6.1 The Underlying Stochastic Dynamics......Page 264
6.2 ItΓ΄'s Rule......Page 266
6.3 Moments and Iteration......Page 269
7 The 1 Term......Page 272
References......Page 274