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Mathematical control theory of coupled PDEs / Irena Lasiecka.

By: Material type: TextTextSeries: CBMS-NSF regional conference series in applied mathematics ; 75.Publisher: Philadelphia : Society for Industrial and Applied Mathematics, [2002]Copyright date: ©2002Description: xii, 242 pages ; 25 cmContent type:
  • text
Media type:
  • unmediated
Carrier type:
  • volume
ISBN:
  • 0898714869
  • 9780898714869
Subject(s): DDC classification:
  • 629.8312 21
LOC classification:
  • QA402.3 .L333 2002
Contents:
Preface -- 1. Introduction -- 1.1. Control Theory of Dynamical PDEs -- 1.1.1. Finite- versus infinite-dimensional control theory -- 1.1.2. Boundary/point control problems for single PDEs -- 1.1.3. Boundary/point control problems for systems of coupled PDEs -- 1.2. Goal of the Lectures -- 2. Well-Posedness of Second-Order Nonlinear Equations with Boundary Damping -- 2.1. Orientation -- 2.2. Abstract Model -- 2.3. Existence and Uniqueness: Statement of Main Results -- 2.4. Nonlinear Plates: von Karman Equations -- 2.4.1. Case [gamma] > 0 -- 2.4.2. Case [gamma] = 0 -- 2.5. Semilinear Wave Equation -- 2.6. Nonlinear Structural Acoustic Model -- 2.7. Full von Karman Systems -- 2.7.1. Model -- 2.7.2. Formulation of the results: Case [gamma] = 0 -- 2.7.3. Formulation of the results: Case [gamma] > 0 -- 2.8. Comments and Open Problems -- 3. Uniform Stabilizability of Nonlinear Waves and Plates -- 3.1. Orientation -- 3.2. Abstract Stabilization Inequalities -- 3.3. Semilinear Wave Equation with Nonlinear Boundary Damping -- 3.3.1. Formulation of the results -- 3.3.2. Regularization -- 3.3.3. Preliminary PDE inequalities -- 3.3.4. Absorption of the lower-order terms -- 3.3.5. Completion of the proof of the main theorem -- 3.4. Nonlinear Plate Equations -- 3.4.1. Modified von Karman equations -- 3.4.2. Full von Karman system and dynamic system of elasticity -- 3.4.3. Nonlinear plates with thermoelasticity -- 3.5. Comments and Open Problems -- 4. Uniform Stability of Structural Acoustic Models -- 4.1. Orientation -- 4.2. Internal Damping on the Wall -- 4.3. Boundary Damping on the Wall -- 4.3.1. Model -- 4.3.2. Formulation of the results -- 4.3.3. Preliminary multipliers estimates -- 4.3.4. Microanalysis estimate for the traces of solutions of Euler-Bernoulli equations and wave equations -- 4.3.5. Observability estimates for the structural acoustic problem -- 4.3.6. Completion of the proof of Theorem 4.3.1 -- 4.4. Thermal Damping -- 4.4.1. Model -- 4.4.2. Statement of main results -- 4.4.3. Sharp trace regularity results -- 4.4.4. Uniform stabilization: Proof of Theorem 4.4.2 -- 4.4.5. Wave equation -- 4.4.6. Uniform stability analysis for the coupled system -- 4.5. Comments and Open Problems -- 5. Structural Acoustic Control Problems: Semigroup and PDE Models -- 5.1. Orientation -- 5.2. Abstract Setting: Semigroup Formulation -- 5.3. PDE Models Illustrating the Abstract Wall Equation (5.2.2) -- 5.3.1. Plates and beams: Flat[Gamma subscript 0] -- 5.3.2. "Undamped" boundary conditions: g [identical with] 0 in (5.3.10) -- 5.3.3. Boundary feedback: Case g [not equal] 0 in (5.3.10) and related stability -- 5.3.4. Shells: Curved-wall [Gamma subscript 0] -- 5.4. Stability in Linear Structural Acoustic Models -- 5.4.1. Internal damping on the wall -- 5.4.2. Boundary damping on the wall -- 5.5. Comments and Open Problems -- 6. Feedback Noise Control in Structural Acoustic Models: Finite Horizon Problems -- 6.1. Orientation -- 6.2. Optimal Control Problem -- 6.3. Formulation of the Results -- 6.3.1. Hyperbolic-parabolic coupling -- 6.3.2. Hyperbolic-hyperbolic coupling: General case -- 6.3.3. Hyperbolic-hyperbolic coupling: Special case of the Kirchhoff plate with point control -- 6.4. Abstract Optimal Control Problem: General Theory -- 6.4.1. Formulation of the abstract control problem -- 6.4.2. Characterization of the optimal control -- 6.4.3. Additional properties under the hyperbolic regularity assumption -- 6.4.4. DRE, feedback generator, and regularity of the gains B*P, B*r -- 6.5. Riccati Equations Subject to the Singular Estimate for e[superscript At]B -- 6.5.1. Formulation of the results -- 6.5.2. Proof of Lemma 6.5.1 -- 6.5.3. Proof of Theorem 6.5.1 -- 6.6. Back to Structural Acoustic Problems: Proofs of Theorems 6.3.1 and 6.3.2 -- 6.6.1. Verification of Assumption (6.4.1) -- 6.6.2. Verification of Assumption 6.5.1 -- 6.7. Comments and Open Problems -- 7. Feedback Noise Control in Structural Acoustic Models: Infinite Horizon Problems -- 7.1. Orientation -- 7.2. Optimal Control Problem -- 7.3. Formulation of the Results -- 7.3.1. Hyperbolic-parabolic coupling -- 7.3.2. Hyperbolic-hyperbolic coupling: Abstract results -- 7.3.3. Hyperbolic-hyperbolic coupling: Kirchhoff plate with point control -- 7.4. Abstract Optimal Control Problem: General Theory -- 7.4.1. Formulation of the abstract control problem -- 7.4.2. ARE subject to condition (7.4.15) -- 7.5. ARE Subject to a Singular Estimate for e[superscript At]B -- 7.5.1. Formulation of the results -- 7.5.2. Proof of Theorem 7.5.1 -- 7.6. Back to Structural Acoustic Problems: Proofs of Theorems 7.3.1 and 7.3.2 -- 7.7. Comments and Open Problems -- Bibliography -- Index.
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Includes bibliographical references (pages 225-238) and index.

Preface -- 1. Introduction -- 1.1. Control Theory of Dynamical PDEs -- 1.1.1. Finite- versus infinite-dimensional control theory -- 1.1.2. Boundary/point control problems for single PDEs -- 1.1.3. Boundary/point control problems for systems of coupled PDEs -- 1.2. Goal of the Lectures -- 2. Well-Posedness of Second-Order Nonlinear Equations with Boundary Damping -- 2.1. Orientation -- 2.2. Abstract Model -- 2.3. Existence and Uniqueness: Statement of Main Results -- 2.4. Nonlinear Plates: von Karman Equations -- 2.4.1. Case [gamma] > 0 -- 2.4.2. Case [gamma] = 0 -- 2.5. Semilinear Wave Equation -- 2.6. Nonlinear Structural Acoustic Model -- 2.7. Full von Karman Systems -- 2.7.1. Model -- 2.7.2. Formulation of the results: Case [gamma] = 0 -- 2.7.3. Formulation of the results: Case [gamma] > 0 -- 2.8. Comments and Open Problems -- 3. Uniform Stabilizability of Nonlinear Waves and Plates -- 3.1. Orientation -- 3.2. Abstract Stabilization Inequalities -- 3.3. Semilinear Wave Equation with Nonlinear Boundary Damping -- 3.3.1. Formulation of the results -- 3.3.2. Regularization -- 3.3.3. Preliminary PDE inequalities -- 3.3.4. Absorption of the lower-order terms -- 3.3.5. Completion of the proof of the main theorem -- 3.4. Nonlinear Plate Equations -- 3.4.1. Modified von Karman equations -- 3.4.2. Full von Karman system and dynamic system of elasticity -- 3.4.3. Nonlinear plates with thermoelasticity -- 3.5. Comments and Open Problems -- 4. Uniform Stability of Structural Acoustic Models -- 4.1. Orientation -- 4.2. Internal Damping on the Wall -- 4.3. Boundary Damping on the Wall -- 4.3.1. Model -- 4.3.2. Formulation of the results -- 4.3.3. Preliminary multipliers estimates -- 4.3.4. Microanalysis estimate for the traces of solutions of Euler-Bernoulli equations and wave equations -- 4.3.5. Observability estimates for the structural acoustic problem -- 4.3.6. Completion of the proof of Theorem 4.3.1 -- 4.4. Thermal Damping -- 4.4.1. Model -- 4.4.2. Statement of main results -- 4.4.3. Sharp trace regularity results -- 4.4.4. Uniform stabilization: Proof of Theorem 4.4.2 -- 4.4.5. Wave equation -- 4.4.6. Uniform stability analysis for the coupled system -- 4.5. Comments and Open Problems -- 5. Structural Acoustic Control Problems: Semigroup and PDE Models -- 5.1. Orientation -- 5.2. Abstract Setting: Semigroup Formulation -- 5.3. PDE Models Illustrating the Abstract Wall Equation (5.2.2) -- 5.3.1. Plates and beams: Flat[Gamma subscript 0] -- 5.3.2. "Undamped" boundary conditions: g [identical with] 0 in (5.3.10) -- 5.3.3. Boundary feedback: Case g [not equal] 0 in (5.3.10) and related stability -- 5.3.4. Shells: Curved-wall [Gamma subscript 0] -- 5.4. Stability in Linear Structural Acoustic Models -- 5.4.1. Internal damping on the wall -- 5.4.2. Boundary damping on the wall -- 5.5. Comments and Open Problems -- 6. Feedback Noise Control in Structural Acoustic Models: Finite Horizon Problems -- 6.1. Orientation -- 6.2. Optimal Control Problem -- 6.3. Formulation of the Results -- 6.3.1. Hyperbolic-parabolic coupling -- 6.3.2. Hyperbolic-hyperbolic coupling: General case -- 6.3.3. Hyperbolic-hyperbolic coupling: Special case of the Kirchhoff plate with point control -- 6.4. Abstract Optimal Control Problem: General Theory -- 6.4.1. Formulation of the abstract control problem -- 6.4.2. Characterization of the optimal control -- 6.4.3. Additional properties under the hyperbolic regularity assumption -- 6.4.4. DRE, feedback generator, and regularity of the gains B*P, B*r -- 6.5. Riccati Equations Subject to the Singular Estimate for e[superscript At]B -- 6.5.1. Formulation of the results -- 6.5.2. Proof of Lemma 6.5.1 -- 6.5.3. Proof of Theorem 6.5.1 -- 6.6. Back to Structural Acoustic Problems: Proofs of Theorems 6.3.1 and 6.3.2 -- 6.6.1. Verification of Assumption (6.4.1) -- 6.6.2. Verification of Assumption 6.5.1 -- 6.7. Comments and Open Problems -- 7. Feedback Noise Control in Structural Acoustic Models: Infinite Horizon Problems -- 7.1. Orientation -- 7.2. Optimal Control Problem -- 7.3. Formulation of the Results -- 7.3.1. Hyperbolic-parabolic coupling -- 7.3.2. Hyperbolic-hyperbolic coupling: Abstract results -- 7.3.3. Hyperbolic-hyperbolic coupling: Kirchhoff plate with point control -- 7.4. Abstract Optimal Control Problem: General Theory -- 7.4.1. Formulation of the abstract control problem -- 7.4.2. ARE subject to condition (7.4.15) -- 7.5. ARE Subject to a Singular Estimate for e[superscript At]B -- 7.5.1. Formulation of the results -- 7.5.2. Proof of Theorem 7.5.1 -- 7.6. Back to Structural Acoustic Problems: Proofs of Theorems 7.3.1 and 7.3.2 -- 7.7. Comments and Open Problems -- Bibliography -- Index.

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