Trends in control theory and partial differential equations

✍ Scribed by Alabau-Boussouira F (ed.)

• Publisher: Springer
• Year: 2019
• Tongue: English
• Leaves: 285
• Series: INdAM 32
• Category: Library

✦ Table of Contents

Preface......Page 6
Contents......Page 9
About the Editors......Page 11
1 Introduction and Statement of the Results......Page 13
1.1 Basic Objects......Page 18
2.1 Preliminaries......Page 23
2.2 Proof of Theorem 1.4......Page 25
References......Page 28
1 Introduction......Page 29
2 Preliminaries......Page 31
3 Uniform Distance Estimates......Page 32
4 Uniform IPC for Functional Set Constraints......Page 39
5 Lipschitz Continuity for a Class of Value Functions......Page 41
6 Applications to the Relaxation Problem......Page 45
References......Page 49
1 Introduction......Page 51
2 Existence of Minimizers in Herglotz' Variational Principle......Page 54
3.1 Lipschitz Estimate of Minimizers......Page 62
3.2 Regularity of Minimizers-Herglotz Equations–Lie Equations......Page 69
4.1 Equivalence of Herglotz' Variational Principle and the Implicit Variational Principle......Page 72
4.2 Herglotz' Generalized Variational Principle on Manifolds......Page 74
4.3 Further Remarks......Page 77
References......Page 78
1 Introduction......Page 80
2.1 Some Preliminary Lemmas......Page 87
2.2 Derivation of the Carleman Estimate......Page 88
3.1 Energy Estimates......Page 92
3.2 The Proof......Page 94
References......Page 97
1 Introduction......Page 99
2 The Weak Maximum Principle in Bounded Domains: A Numerical Criterion......Page 100
3 Unbounded Domains and Uniform Ellipticity......Page 104
4 One-Directional Elliptic Operators on Special Unbounded Domains......Page 107
5 An Approximation of the Principal Eigenvalue......Page 110
References......Page 113
On the Convergence of Open Loop Nash Equilibria in Mean Field Games with a Local Coupling......Page 115
1.1 Notation......Page 117
1.2 Assumption......Page 118
1.3 Regularity Estimates......Page 119
2 Open Loop Nash Equilibria......Page 120
3.1 Estimates Between vN,i and uN......Page 123
3.2 Putting the Estimates Together......Page 130
References......Page 132
1 Introduction......Page 133
2 Carleman Inequalities and Null Controllability......Page 136
3 Local Null Controllability of the b-Equations......Page 139
4 Controllability in the Limit......Page 141
5.1 Exponential Decay and Large Time Null Controllability for the Burgers-α Equation......Page 142
5.3 The Situation in Higher Spatial Dimensions......Page 145
References......Page 146
1-d Wave Equations Coupled via Viscoelastic Springs and Masses: Boundary Controllability of a Quasilinear and Exponential Stabilizability of a Linear Model......Page 149
1 Introduction......Page 150
2.1 Well-Posedness......Page 154
2.2 Dissipativity of the Nonlinear Model......Page 157
3 Exact Boundary Controllability for the Kelvin-Type Viscoelastic Coupling......Page 158
4 Exponential Boundary Stabilization of a Linear Kelvin–Voigt-Model......Page 159
5 Conclusion and Outlook......Page 164
References......Page 165
1 Introduction......Page 167
2 Preliminaries......Page 170
3 Existence and Uniqueness of Mild and Strong Solutions......Page 173
4 Hidden Regularity Results......Page 180
References......Page 190
1 Introduction......Page 191
2 A Dual Approach to Lyapunov's Theorem......Page 193
References......Page 204
Controllability Under Positivity Constraints of Multi-d Wave Equations......Page 205
1 Introduction......Page 206
1.1 Internal Control......Page 207
1.2 Boundary Control......Page 210
2 Abstract Results......Page 214
2.1 Steady State Controllability......Page 216
2.2 Controllability Between Trajectories......Page 221
3.1 Proof of Theorem 1......Page 229
3.3 Internal Controllability From a Neighborhood of the Boundary......Page 230
4 Boundary Control: Proof of Theorems 3, 4 and 5......Page 231
4.1 Proof of Theorem 3......Page 232
4.3 State Constraints. Proof of Theorem 5......Page 233
5 The One Dimensional Wave Equation......Page 235
6 Conclusions and Open Problems......Page 237
References......Page 241
1 Introduction......Page 243
2 Preliminary Properties......Page 246
3 Convergence to Consensus......Page 248
4 The Case of Free-Will Leader......Page 254
References......Page 261
1.1 Introduction......Page 264
1.3 Main Result......Page 266
1.4 Other Results......Page 267
1.5 Perpectives......Page 268
2.1 Functional Framework......Page 269
2.2 The Unperturbed Operator......Page 270
2.3 Perturbed Operators......Page 271
2.4 Maximum Principle......Page 272
2.5 Specific Perturbed Operator......Page 273
3.1 Step 1......Page 276
3.2 Step 2......Page 277
3.3 Step 3......Page 278
4.1 Proof of Theorem 1.2......Page 279
4.2 Proof of Theorem 1.3......Page 280
References......Page 283