Control Theory for Partial Differential Equations: Volume 2, Abstract Hyperbolic-Like Systems Over a Finite Time Horizon: Continuous and Approximation

✍ Scribed by Lasiecka, Irena;Triggiani, Roberto

• Publisher: Cambridge University Press
• Year: 2000
• Tongue: English
• Leaves: 452
• Series: Encyclopedia of mathematics and its applications 75
• Category: Library

✦ Synopsis

Volume II focuses on the optimal control problem over a finite time interval for hyperbolic dynamical systems. The chapters consider some abstract models, each motivated by a particular canonical hyperbolic dynamics, and present numerous new results.

✦ Table of Contents

Cover......Page 1
Half Title......Page 3
Series Page......Page 5
Dedication......Page 6
Title......Page 7
Copyright......Page 8
Contents......Page 9
Preface......Page 17
Acknowledgments for the First Two Volumes......Page 22
7.1 Mathematical Setting and Standing Assumptions......Page 25
7.2 Regularity of Land L * on [0, T]......Page 28
7.3 A Lifting Regularity Property When eAt Is a Group......Page 31
7.4.1 Direct Statement; Direct Proof......Page 33
7.4.2 Dual Statement; Dual Proof......Page 37
7.5 Generation and Abstract Trace Regularity under Unbounded Perturbation......Page 40
7.6.1 Mathematical Setting and Assumptions......Page 43
7.6.2 Main Regularity Results......Page 44
7.6.3 Proof of Theorem 7.6.2.2: Dual Statement (7.6.2.6)......Page 45
7.7.1 Wave Equation with Boundary Damping in the Neumann Be......Page 47
7.7.2 Wave Equation with Boundary Damping in the Dirichlet BC......Page 49
References and Bibliography......Page 51
8 Optimal Quadratic Cost Problem Over a Preassigned Finite Time Interval: The Case Where the Input → Solution Map Is Unbounded, but the Input → Observation Map Is Bounded......Page 53
8.1 Mathematical Setting and Formulation of the Problem......Page 55
8.2.1 The General Case: Theorem 8.2.1.1, Theorem 8.2.1.2, and Theorem 8.2.1.3......Page 59
8.2.2 The Regular Case: Theorem 8.2.2.1......Page 65
8.3.1 Explicit Representation Formulas for the Optimal Pair {uo, yo} under (h.1), (h.3)......Page 67
8.3.2 Estimates on uo ( • ,t; x) and Ryo ( . ,t; x). The Operator Φ ( . , . )......Page 71
8.3.3 Definition of P(t) and Preliminary Properties......Page 75
8.3.4 P(t) Solves the Differential Riccati Equation (8.2.1.32)......Page 79
8.3.5 Differential and Integral Riccati Equations......Page 84
8.3.6 The IRE without Passing through the DRE......Page 87
8.3.7 Uniqueness......Page 89
8.3.8 Proof of Theorem 8.2.1.3......Page 93
8.4 A Second Direct Proof of Theorem 8.2.1.2: From the Well-Posedness of the IRE to the Control Problem. Dynamic Programming......Page 94
8.4.1 Existence and Uniqueness: Preliminaries......Page 95
8.4.2 Unique Local Solution to Eqn. (8.4.1.5)for Q(t, s)......Page 98
8.4.3 Unique Local Solution to Eqn. (8.4.1. 7) for V(t). Global Solution P(t) under (h.1), (h.2)......Page 100
8.4.4 Global A Priori Estimates for V and Q. Global Solution P(t) under (H. 1), (H.2), and (H.3)......Page 102
8.4.5 Recovering the Optimal Control Problem under (H.I), (H.2), and (H.3) for (h.I) and (h.2)]......Page 109
8.5.1 A Preliminary Lemma......Page 113
8.5.2 Completion of the Proof of Theorem 8.2.2.1......Page 114
8.5.3 An Auxiliary Lemma......Page 115
8.6.1 Problem Formulation and Abstract Setting......Page 116
8.6.2 Specialization of Theorems 8.2.1.1 and 8.2.1.2 to the Hyperbolic Problem: Theorem 8.6.2.1......Page 121
8.6.3 Proof of Theorem 8.6.2.1: Verification of Assumptions (H.l), (H.2), and (H.3)......Page 122
8.6.4 Specialization of Theorem 8.2.2.1 to the Hyperbolic Problem: Verification of Assumptions (H.4) and (H.S)......Page 123
8.7.1 Introduction. Summary of Results......Page 125
8.7.2 The Optimal Control Problem and Its Solution......Page 126
8.7.3 Abstract Model......Page 128
8.7.4 Analysis of the Corresponding, Integral or Differential, Riccati Equation......Page 131
8.7.5 Further Abstract Properties......Page 133
8A Interior and Boundary Regularity of Mixed Problems for Second-Order Hyperbolic Equations with Neumann-Type BC......Page 135
Some Basic Regularity Results......Page 136
Regularity Results with Data {wo, w., f} Less Regular than L2......Page 140
Regularity Theory for Second-Order Mixed Hyperbolic Equations with Neumann Boundary Datum......Page 141
References and Bibliography......Page 143
9.1 Mathematical Setting and Formulation of the Problem......Page 145
9.2 Statement of Main Result: Theorems 9.2.1, 9.2.2, and 9.2.3......Page 152
9.3 Proofs of Theorem 9.2.1 and Theorem 9.2.2 (by the Variational Approach and by the Direct Approach). Proof of Theorem 9.2.3......Page 156
9.3.1 A First Proofby the Variational Method. Theorem 9.2.1: Explicit Representation Formulas for the Optimal Pair {uo, yo} under (A.l)......Page 157
9.3.2 Theorem 9.2.1: Estimates on uo( • , t; x) and yo( • , t; x) under (A.1). The Operator Φ ( • , • )......Page 159
9.3.3 Definition of P(t) and Preliminary Properties......Page 164
9.3.4 Theorem 9.2.2: P(t) Solves the Differential Riccati Equation (9.2.21) under (A.l), (A.2), and (A.3)......Page 172
9.3.5 Merger with the Proof of Chapter 8: Uniqueness and the IRE......Page 179
9.3.6 A Second Direct Proof of Theorem 9.2.2: From the Well-Posedness of the IRE to the Control Problem. Dynamic Programming......Page 181
9.3.6.1 Local Well-Posedness of the IRE (9.2.22)......Page 182
9.3.7 Proof of Theorem 9.2.3......Page 192
9.4 Isomorphism of P(t), 0 ≤ t < T, and Exact Controllability of {A, R} on [0, T - t] When G = 0......Page 195
9.5.1 Introduction......Page 199
9.5.2 Regularization of R. Statement of Main Result: Theorem 9.5.2.2......Page 200
9.5.3 Proof of Theorem 9.5.2.2......Page 203
9.6.1 Motivation. Orientation.......Page 205
9.6.2 Dual Optimal Control Problem; Dual Differential Riccati Equation, When A is a Group Generator......Page 206
9.6.3 Proof of Theorem 9.6.2.2......Page 209
9.6.4 Proof of Theorem 9.6.2.3......Page 211
9.6.5 Proof of Theorem 9.6.2.4......Page 215
9.6.6 A Transition Property of Q( . )......Page 216
9.6.7 1somorphism of Qr(t), 0 ≤ t ≤ T and Exact Controllability of {A, B} on [0, T - t]......Page 217
9.7 Optimal Control Problem with Bounded Control Operator and Unbounded Observation Operator......Page 219
9.8.1 Applications: Wave Equation with Interior Point Control and Dirichlet Boundary Conditions......Page 222
9.8.2 Applications: Wave Equation with Interior Point Control and Boundary Observation......Page 227
9.8.3 Applications: Kirchhoff Equation with Interior Point Control......Page 233
9.8.4 One-Dimensional Wave Equation with Neumann Boundary Control......Page 237
9.9 Proof of Regularity Results Needed in Section 9.8......Page 241
9.9.1 Proof of Theorem 9.S.1.1......Page 242
9.9.2 Proof of Theorem 9.8.2.1......Page 250
9.9.3 Proof of Theorem 9.8.3.1......Page 254
9.9.4 Proof of Theorem 9.8.4.1......Page 262
9.10.1 Statement of Problem for dim Ω = 2 Main Results......Page 264
9.10.2 Abstract Model of the Original Problem (9.10.1.1) in {z, v} (Hinged BC). Theorem 9.10.1.1......Page 268
9.10.3 Proof of Theorem 9.10.1.2: Regularity of Lin (9.10.1.5)......Page 271
9.10.4 A Corresponding Damped Model......Page 277
9.11.1 Statement 0/ Problem/or dim n = 2. Main Results......Page 281
9.11.2 Abstract Model of Original Problem (9.11.1.1) in {z, v}. Theorem 9.11.1.1......Page 283
9.11.3 Proof of Theorem 9.11.1.2: Abstract Trace Condition (9.11.1.9)......Page 286
9A Proof of (9.9.1.16) in Lemma 9.9.1.1......Page 288
9B Proof of (9.9.3.14) in Lemma 9.9.3.1......Page 290
Notes on Chapter 9......Page 293
References and Bibliography......Page 296
10 Differential Riccati Equations under Slightly Smoothing Observation Operator. Applications to Hyperbolic and Petrowski-Type PDEs. Regularity Theory......Page 299
10.1 Mathematical Setting and Problem Statement......Page 300
10.2 Statement of the Main Results......Page 306
10.3.1 Bounded Inversion of[Is + LsLs R R] on the Space y½-∂[s, T], Uniformly in s. Proof of Theorem 10.2.1......Page 308
10.3.2 Proof of Theorem 10.2.2......Page 313
10.4.1 Bounded Inversion of[ls + LsLs R R] on the Space L2(s, T; [Y+∂]'). Consequences on Φ (t, s )......Page 316
10.4.2 Derivation of the Differential and Integral Riccati Equations......Page 319
10.5 Application: Second-Order Hyperbolic Equations with Dirichlet Boundary Control. Regularity Theory......Page 322
10.5.1 Problem Formulation......Page 323
10.5.2 Main Results......Page 324
10.5.3 Regularity Theory for Problem (10.5.1.3) with u € L2(S)......Page 325
10.5.4 Abstract Setting for Problem (10.5.1.3)......Page 327
10.5.5 Verification of Assumption (H.1) = (10.1.6)......Page 330
10.5.6 Selection of the Spaces VΘ and YΘ in (10.1.10)......Page 333
10.5.7 Verification of Assumption (H.2) = (10.1.14)......Page 334
10.5.S Verification of Assumption (H.3) = (10.1.15)......Page 335
10.5.9 Verification of Assumptions (H.4) = (10.1.18) through (H. 7) = (10.1.21)......Page 336
10.5.10 Proof of Theorem 10.5.3.1......Page 338
10.5.11 Proof of Theorem 10.5.3.2......Page 343
10.5.12 Proof of Theorem 10.5.7.1......Page 347
10.5.13 Proof of Theorem 10.5.S.1......Page 349
10.6.1 Problem Formulation......Page 352
10.6.2 Main Results......Page 354
10.6.3 Regularity Theory for Problem (10.6.1.5) with u € L2(Σ)......Page 355
10.6.4 Abstract Setting for Problem (10.6.1.5)......Page 356
10.6.5 Verification of Assumptions (H.1) = (10.1.6) through (H.3) = (10.1.15)......Page 358
10.6.6 Selection of the Spaces UΘ and YΘ in (10.1.10)......Page 361
10.6.8 Verification of Assumption (H.3) = (10.1.15)......Page 362
10.6.9 Verification of Assumptions (H.4) = (10.1.18) through (H. 7) = (10.1.21)......Page 365
10.6.10 Proof of Lemma 10.6.4.1: Existence and Regularity of the Map Dλ......Page 366
10.6.11 Proof of Lemma 10.6.4.2......Page 368
10.7 Application: Kirchoff Equation with One Boundary Control. Regularity Theory......Page 369
10.7.2 Main Results......Page 370
10.7.3 Regularity Theory for Problem (10.7.1.1) with u € L2(S)......Page 372
10.7.4 Abstract Setting for Problem (10.7.1.1)......Page 374
10.7.5 Verification of Assumption (H.l) = (10.1.6)......Page 378
10.7.6 Selection of the Spaces UΘ and YΘ in (10.1.10)......Page 380
10.7.7 Verification of Assumption (H.2) = (10.1.14)......Page 381
10.7.S Verification of Assumption (H.3) = (10.1.15)......Page 382
10.7.9 Verification of Assumptions (H.4) = (10.1.18) through (H.7) = (10.1.21)......Page 383
10.7.10 Proof of Theorem 10.7.3.1......Page 385
10.7.11 Proof of Theorem 10.7.3.2......Page 389
10.7.12 Proof of Theorem 10.7.7.1......Page 392
10.7.13.1 A Preliminary Trace Result......Page 394
10.8.1 Problem Formulation......Page 399
10.8.2 Main Results......Page 400
10.8.3 Regularity Theory for Problem (10.8.1.1) with u € L2(Σ)......Page 402
10.8.4 Abstract Setting for Problem (10.8.1.1)......Page 403
10.8.5 Verification of Assumption (H.l) = (10.1.6)......Page 406
10.8.6 Selection of the Spaces UΘ and YΘ in (10.1.10)......Page 407
10.8.7 Verification of Assumptions (H.2) = (10.1.14)......Page 409
10.8.8 Verification of Assumption (H.3) = (10.1.15)......Page 410
10.8.10 Proof of Theorem 10.8.3.1......Page 412
10.8.11 ProofofTheorem 10.8.3.2......Page 414
10.8.12 Proof of Theorem 10.8.7.1......Page 416
10.8.13.1 A Preliminary Trace Result......Page 417
10.8.13.2 Completion of the Proof of Theorem 10.8.8.1......Page 421
10.9.2 Main Results......Page 422
10.9.3 Regularity Theory for Problem (10.9.1.1) with u € L2(Σ)......Page 424
10.9.4 Abstract Setting for Problem (10.9.1.1)......Page 425
10.9.5 Verification of Assumption (H.l) = (10.1.6)......Page 427
10.9.6 Selection of the Spaces UT and YT in (10.1.10)......Page 428
10.9.7 Verification of Assumptions (H.2) = (10.1.14)......Page 429
10.9.S Verification of Assumption (H.3) = (10.1.15)......Page 430
10.9.10 Proof of Theorem 10.9.3.1......Page 431
10.9.11 Proof of Theorem 10.9.3.2......Page 434
10.9.12 Proof of Theorem 10.9.7.1......Page 436
10.9.13 Proof of Theorem 10.9.S.1......Page 437
Notes on Chapter 10......Page 439
Hyperbolic/Platelike Case......Page 444
References and Bibliography......Page 445
Index......Page 449