Control Theory of Infinite-Dimensional Systems

✍ Scribed by Kerner, Joachim, Laasri, Hafida, Mugnolo, Delio

• Publisher: Springer
• Year: 2020
• Tongue: English
• Leaves: 201
• Series: Linear Operators and Linear Systems, 277
• Edition: 1
• Category: Library

✦ Synopsis

This book presents novel results by participants of the conference “Control theory of infinite-dimensional systems” that took place in January 2018 at the FernUniversität in Hagen. Topics include well-posedness, controllability, optimal control problems as well as stability of linear and nonlinear systems, and are covered by world-leading experts in these areas.

A distinguishing feature of the contributions in this volume is the particular combination of researchers from different fields in mathematics working in an interdisciplinary fashion on joint projects in mathematical system theory. More explicitly, the fields of partial differential equations, semigroup theory, mathematical physics, graph and network theory as well as numerical analysis are all well-represented.

✦ Table of Contents

Preface......Page 6
Contents......Page 7
1. Introduction......Page 8
2.1. Notation......Page 11
2.2. Basic definitions......Page 14
3. Examples and Previous Results......Page 16
4. Port-Hamiltonian Systems: Networks......Page 22
5. Stability Properties of Hybrid Multi-PHS-control systems......Page 30
6. Networks of Hybrid PH-ODE Systems......Page 35
7. Applications......Page 38
8. Conclusion and Open Problems......Page 51
9. Appendix: Some technical results on the Euler–Bernoulli Beam......Page 52
References......Page 57
1.1. Operator convergence in varying Hilbert spaces......Page 60
1.2. Metrics on sets of operators acting in different Hilbert spaces......Page 61
2.1. A spectral distance for operators acting in different Hilbert spaces......Page 62
2.2. Quasi-unitary equivalence......Page 66
2.3. Consequences of quasi-unitary equivalence......Page 68
2.4. A distance arising from quasi-unitary equivalence......Page 69
3.1. Metric graphs......Page 70
3.2. Thin branched manifolds (“fat graphs”)......Page 71
3.3. Convergence of the Laplacian on thin branched manifolds......Page 73
References......Page 78
1. Introduction......Page 80
2. Some basic facts about admissibility and infinite-time admissibility......Page 81
3.2. Infinite-time admissibility under compact perturbations......Page 83
4.1. Characterization of infinite-time admissibility......Page 86
4.2. Infinite-time admissibility under compact perturbations......Page 87
References......Page 88
1. Introduction......Page 90
1.1. ISS for parabolic semilinear systems — what is known......Page 95
1.2. Notation......Page 96
2. A recap on ISS for linear boundary control systems......Page 97
3. A primer on semilinear boundary control systems......Page 113
4. Concluding remarks and outlook......Page 119
References......Page 120
1. Introduction......Page 124
2. Scale-free spectral inequalities based on complex analysis......Page 126
2.1. Earlier literature and historical development: Equivalent norms on subspaces......Page 127
2.2. Current state-of-the-art......Page 129
3. Scale-free spectral inequalities based on Carleman estimates......Page 135
3.1. Development of scale-free unique continuation estimates applicable to Schrödinger operators with random potential......Page 137
3.2. Current state-of-the-art......Page 140
4. From uncertainty to control......Page 142
5. Null-controllability of the heat and Schrödinger semigroups......Page 152
6. Convergence of solutions along exhausting cubes......Page 157
6.2. Continuous dependence on inhomogeneity......Page 158
6.3. Construction of controls via exhaustion of the domain......Page 159
References......Page 161
1. Introduction......Page 165
2. Preliminaries......Page 169
3. Two scales of Hilbert spaces associated with a closed operator......Page 173
4. The Hamiltonian......Page 177
5. Bisectorial Hamiltonians......Page 181
6. Graph and angular subspaces......Page 187
7. Symmetries of the angular operators......Page 194
8. The Riccati equation......Page 197
References......Page 200