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Tables of Contents for Mathematical Control Theory of Coupled Pdes
Chapter/Section Title
Page #
Page Count
Preface
xi
Introduction
1
6
Control Theory of Dynamical PDEs
1
3
Finite- versus infinite-dimensional control theory
2
1
Boundary/point control problems for single PDEs
2
1
Boundary/point control problems for systems of coupled PDEs
3
1
Goal of the Lectures
4
3
Well-Posedness of Second-Order Nonlinear Equations with Boundary Damping
7
32
Orientation
7
1
Abstract Model
8
1
Existence and Uniqueness: Statement of Main Results
9
4
Nonlinear Plates: von Karman Equations
13
8
Case γ > 0
15
4
Case γ = 0
19
2
Semilinear Wave Equation
21
4
Nonlinear Structural Acoustic Model
25
3
Full von Karman Systems
28
9
Model
28
4
Formulation of the results: Case γ = 0
32
2
Formulation of the results: Case γ > 0
34
3
Comments and Open Problems
37
2
Uniform Stabilizability of Nonlinear Waves and Plates
39
46
Orientation
39
2
Abstract Stabilization Inequalities
41
4
Semilinear Wave Equation with Nonlinear Boundary Damping
45
22
Formulation of the results
47
4
Regularization
51
5
Preliminary PDE inequalities
56
5
Absorption of the lower-order terms
61
5
Completion of the proof of the main theorem
66
1
Nonlinear Plate Equations
67
15
Modified von Karman equations
67
5
Full von Karman system and dynamic system of elasticity
72
5
Nonlinear plates with thermoelasticity
77
5
Comments and Open Problems
82
3
Uniform Stability of Structural Acoustic Models
85
48
Orientation
85
1
Internal Damping on the Wall
86
4
Boundary Damping on the Wall
90
20
Model
90
2
Formulation of the results
92
5
Preliminary multipliers estimates
97
4
Microanalysis estimate for the traces of solutions of Euler-Bernoulli equations and wave equations
101
3
Observability estimates for the structural acoustic problem
104
5
Completion of the proof of Theorem 4.3.1
109
1
Thermal Damping
110
20
Model
110
2
Statement of main results
112
3
Sharp trace regularity results
115
2
Uniform stabilization: Proof of Theorem 4.4.2
117
7
Wave equation
124
2
Uniform stability analysis for the coupled system
126
4
Comments and Open Problems
130
3
Structural Acoustic Control Problems: Semigroup and PDE Models
133
30
Orientation
133
2
Abstract Setting: Semigroup Formulation
135
5
PDE Models Illustrating the Abstract Wall Equation (5.2.2)
140
15
Plates and beams: Flat Λ0
140
3
``Undamped'' boundary conditions: g ≠ 0 in (5.3.10)
143
2
Boundary feedback: Case g ≠ 0 in (5.3.10) and related stability
145
6
Shells: Curved-wall Λ0
151
4
Stability in Linear Structural Acoustic Models
155
5
Internal damping on the wall
156
2
Boundary damping on the wall
158
2
Comments and Open Problems
160
3
Feedback Noise Control in Structural Acoustic Models: Finite Horizon Problems
163
40
Orientation
163
2
Optimal Control Problem
165
2
Formulation of the Results
167
7
Hyperbolic-parabolic coupling
167
1
Hyperbolic-hyperbolic coupling: General case
168
1
Hyperbolic-hyperbolic coupling: Special case of the Kirchhoff plate with point control
169
5
Abstract Optimal Control Problem: General Theory
174
6
Formulation of the abstract control problem
174
1
Characterization of the optimal control
175
2
Additional properties under the hyperbolic regularity assumption
177
2
DRE, feedback generator, and regularity of the gains B* P, B*r
179
1
Riccati Equations Subject to the Singular Estimate for eAtB
180
10
Formulation of the results
180
1
Proof of Lemma 6.5.1
181
6
Proof of Theorem 6.5.1
187
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 6.5.1
193
8
Comments and Open Problems
201
2
Feedback Noise Control in Structural Acoustic Models: Infinite Horizon Problems
203
22
Orientation
203
2
Optimal Control Problem
205
1
Formulation of the Results
206
6
Hyperbolic-parabolic coupling
206
2
Hyperbolic-hyperbolic coupling: Abstract results
208
1
Hyperbolic-hyperbolic coupling: Kirchhoff plate with point control
209
3
Abstract Optimal Control Problem: General Theory
212
2
Formulation of the abstract control problem
212
1
ARE subject to condition (7.4.15)
213
1
ARE Subject to a Singular Estimate for eAtB
214
8
Formulation of the results
214
1
Proof of Theorem 7.5.1
215
7
Back to Structural Acoustic Problems: Proofs of Theorems 7.3.1 and 7.3.2
222
1
Comments and Open Problems
222
3
Bibliography
225
14
Index
239
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