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Tables of Contents for Mathematical Control Theory of Coupled Pdes
Control Theory of Dynamical PDEs
1
3
Finite- versus infinite-dimensional control theory2
1
Boundary/point control problems for single PDEs2
1
Boundary/point control problems for systems of coupled PDEs3
1
Well-Posedness of Second-Order Nonlinear Equations with Boundary Damping
7
32
Existence and Uniqueness: Statement of Main Results
9
4
Nonlinear Plates: von Karman Equations
13
8
Case γ > 015
4
Case γ = 019
2
Semilinear Wave Equation
21
4
Nonlinear Structural Acoustic Model
25
3
Full von Karman Systems
28
9
Model28
4
Formulation of the results: Case γ = 032
2
Formulation of the results: Case γ > 034
3
Comments and Open Problems
37
2
Uniform Stabilizability of Nonlinear Waves and Plates
39
46
Abstract Stabilization Inequalities
41
4
Semilinear Wave Equation with Nonlinear Boundary Damping
45
22
Formulation of the results47
4
Regularization51
5
Preliminary PDE inequalities56
5
Absorption of the lower-order terms61
5
Completion of the proof of the main theorem66
1
Nonlinear Plate Equations
67
15
Modified von Karman equations67
5
Full von Karman system and dynamic system of elasticity72
5
Nonlinear plates with thermoelasticity77
5
Comments and Open Problems
82
3
Uniform Stability of Structural Acoustic Models
85
48
Internal Damping on the Wall
86
4
Boundary Damping on the Wall
90
20
Model90
2
Formulation of the results92
5
Preliminary multipliers estimates97
4
Microanalysis estimate for the traces of solutions of Euler-Bernoulli equations and wave equations101
3
Observability estimates for the structural acoustic problem104
5
Completion of the proof of Theorem 4.3.1109
1
Model110
2
Statement of main results112
3
Sharp trace regularity results115
2
Uniform stabilization: Proof of Theorem 4.4.2117
7
Wave equation124
2
Uniform stability analysis for the coupled system126
4
Comments and Open Problems
130
3
Structural Acoustic Control Problems: Semigroup and PDE Models
133
30
Abstract Setting: Semigroup Formulation
135
5
PDE Models Illustrating the Abstract Wall Equation (5.2.2)
140
15
Plates and beams: Flat Λ0140
3
``Undamped'' boundary conditions: g ≠ 0 in (5.3.10)143
2
Boundary feedback: Case g ≠ 0 in (5.3.10) and related stability145
6
Shells: Curved-wall Λ0151
4
Stability in Linear Structural Acoustic Models
155
5
Internal damping on the wall156
2
Boundary damping on the wall158
2
Comments and Open Problems
160
3
Feedback Noise Control in Structural Acoustic Models: Finite Horizon Problems
163
40
Optimal Control Problem
165
2
Formulation of the Results
167
7
Hyperbolic-parabolic coupling167
1
Hyperbolic-hyperbolic coupling: General case168
1
Hyperbolic-hyperbolic coupling: Special case of the Kirchhoff plate with point control169
5
Abstract Optimal Control Problem: General Theory
174
6
Formulation of the abstract control problem174
1
Characterization of the optimal control175
2
Additional properties under the hyperbolic regularity assumption177
2
DRE, feedback generator, and regularity of the gains B* P, B*r179
1
Riccati Equations Subject to the Singular Estimate for eAtB
180
10
Formulation of the results180
1
Proof of Lemma 6.5.1181
6
Proof of Theorem 6.5.1187
3
Back to Structural Acoustic Problems: Proofs of Theorems 6.3.1 and 6.3.2
190
11
Verification of Assumption (6.4.1)192
1
Verification of Assumption 6.5.1193
8
Comments and Open Problems
201
2
Feedback Noise Control in Structural Acoustic Models: Infinite Horizon Problems
203
22
Optimal Control Problem
205
1
Formulation of the Results
206
6
Hyperbolic-parabolic coupling206
2
Hyperbolic-hyperbolic coupling: Abstract results208
1
Hyperbolic-hyperbolic coupling: Kirchhoff plate with point control209
3
Abstract Optimal Control Problem: General Theory
212
2
Formulation of the abstract control problem212
1
ARE subject to condition (7.4.15)213
1
ARE Subject to a Singular Estimate for eAtB
214
8
Formulation of the results214
1
Proof of Theorem 7.5.1215
7
Back to Structural Acoustic Problems: Proofs of Theorems 7.3.1 and 7.3.2
222
1
Comments and Open Problems
222
3
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