x CONTENTS
Chapter 2. First Optimizations: Multiplicity of the Spectrum and the DISC 237
2.1. The Least Dimension of Controlling Subspaces 237
2.2. Reduction to Bounded Operators 238
2.3. Multiplicity of the Spectrum 241
2.4. The Minimal Dimension of Constrained (Realizable) Control 247
2.5. Exercises and Further Results 254
2.6. Notes and Remarks 262
Chapter 3. Eigenvector Decompositions, Vector Valued Exponentials, and
Squared Optimization 267
3.1. Examples of Parabolic and Hyperbolic Systems 268
3.2. Complete Generators 271
3.3. Riesz Bases and Exact Controllability 274
3.4. Generalized Controllability and Renormalizations 277
3.5. Null Controllability (NCO) 284
3.6. Weak Controllability 289
3.7. Squared (Energy) Optimization 290
3.8. Control at Time r = oc and Interpolation in H2 293
3.9. Notes and Remarks 295
Chapter 4. A Glance at Bases of Exponentials and of Reproducing Kernels 299
4.1. Small Perturbations of Harmonic Frequencies 300
4.2. Bases of Exponentials on the Half-Line 301
4.3. Bases of Exponentials on Finite Intervals 304
4.4. Bases of Reproducing Kernels in Model Spaces 307
4.5. Back to Exponentials 313
4.6. A Levinson Completeness Theorem 316
4.7. Exercises and Further Results 321
4.8. Notes and Remarks 334
Chapter 5. A Brief Introduction to H°° Control 343
5.1. Input-Output Maps and Transfer Functions 343
5.2. Noise Minimization, Feedback Control, and Sensitivity 346
5.3. Remarks on Robust Stabilization 350
5.4. Scattering Type Input-Output, and Hankel Operators 350
5.5. Remarks on Finite Dimensional Systems 355
5.6. Exercises and Further Results 356
5.7. Notes and Remarks 358
Bibliography 361
Author Index 401
Subject Index 411
Symbol Index 435
Errata to Volume 1 439
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