Control Theory of Systems Governed by Partial Differential Equations

Title Page......Page 1
Copyright......Page 2
Contents......Page 3
List of Contributors......Page 5
Preface......Page 7
Introduction......Page 9
1.2 The state equation......Page 15
1.4 Standard results......Page 17
1.5 Particular cases......Page 19
2.1.Statement of the problem......Page 20
2.2 The optimality system......Page 21
2.3 Particular cases......Page 23
2.4 Another example......Page 25
2.5 An example of "parabolic-elliptic" nature......Page 26
3.1 Setting of the problem......Page 27
3.2 Optimality system......Page 28
3.4 The case when Uad = {vlv > 0 a.e. on E}......Page 30
4.1 Setting of the problem......Page 31
4.2 The optimality system......Page 33
5.1 Orientation......Page 34
5.2 Formulation as a control problem......Page 35
5.3 Regularization method......Page 36
1.1 Setting of the problem......Page 39
1.2 A formal computation......Page 40
2.1 Orientation......Page 41
2.3 Transformation by duality......Page 42
2.4 Regularized dual problem and generalized problem......Page 46
3.1 Direct method......Page 48
3.2 Use of duality......Page 50
2.1 Setting of the problem......Page 53
2.2 A convergence theorem......Page 54
2.3 Connection with singular perturbations......Page 57
3.1 A model problem......Page 58
3.2 The homogeneized operator......Page 59
3.3 A convergence theorem......Page 60
1.1 Setting of the problem......Page 65
1.2 Optimality conditions......Page 66
1.3 An example......Page 71
2.2 Statement of the problem......Page 72
2.3 Optimality conditions......Page 73
3.1 General remarks......Page 76
3.2 An example......Page 77
4.1 Setting of the problem......Page 80
4.2 Optimality conditions......Page 81
5.1 Variational inequalities and free surfaces......Page 82
5.2 Optimal control of variational inequalities......Page 85
5.3 Open questions......Page 89
6.1 General remarks......Page 90
6.2 Open questions......Page 91
1. General Remarks......Page 93
2.1 Mixed variational problems......Page 94
2.2 Regularization of mixed variational problems......Page 96
2.3 Optimal control of mixed variational systems......Page 101
2.4 Approximation of the optimal control of mixed variational systems......Page 103
Bibliography......Page 106
Stochastic filtering and control of linear systems: a general theory......Page 113
1. Introduction......Page 127
2. Control Problems for Hyperbolic Systems......Page 130
3. Spectral Determination For Hyperbolic Systems......Page 139
4. Spectral Determination for Certain One-Dimensional Diffusion Processes.......Page 149
5. Remarks on Canonical Equations of Higher Order......Page 154
References......Page 158
1. Introduction.......Page 159
2. Second-order equations in Hilbert space.......Page 162
3. Solution of the controllability problem.......Page 166
4. Existence of optimal controls.......Page 172
5. The maximum principle.......Page 173
6. Generalizations. The maximum principle in other geometries.......Page 178
Footnotes......Page 182
1. Introduction......Page 185
2. Extremal Eigenvalue Problems......Page 186
3. The Shape of the Strongest Tubular Column......Page 197
4. Lyapunov Zones of Stability for Hill's Equation......Page 201
5. A Variational Problem Arising in the Design of Cooling Fins......Page 207
ACKNOWLEDGEMENT......Page 214
References......Page 215
1. INTRODUCTION......Page 217
2. VARIATIONAL INEQUALITIES AND TRANSFORMATIONS OF VARIABLES......Page 218
3. RELAXATION METHODS AND EVOLUTION METHODS......Page 221
4. THE METHODS OF OPTIMUM DESIGN......Page 222
References......Page 232
II. A Survey of State Estimation Algorithms and Applications......Page 239
A. Theoretical Results......Page 240
1. Observability and Measurement Location......Page 241
2. State Estimators Available......Page 243
B. AN OVERVIEW OF APPLICATIONS......Page 247
1. Heat Conduction Processes......Page 249
2. Chemical Reactors......Page 252
4. Problems with Moving Boundaries......Page 255
III. A Real Time Case Study:......Page 256
IY. Concluding Remarks......Page 265
References......Page 266
2. Introduction......Page 273
3. Statement of the problem......Page 274
4. Gelder's algorithm for subsonic flow......Page 275
5. Formulation via optimal control......Page 277
6. Discretization and numerical solution......Page 280
8. Conclusion......Page 283
References......Page 286