Mathematical Control Theory of Coupled Systems of Partial Differential Equations (CBMS-NSF Regional Conference Series in Applied Mathematics)

Mathematical control theory for a single partial differential equation (PDE) has dominated the research literature for quite a while: new, complex, and challenging issues have recently arisen in the context of coupled, or interconnected, PDE systems. This has led to a rapidly growing interest, and many unanswered questions, within the PDE community. By concentrating on systems of hyperbolic and parabolic coupled PDEs that are nonlinear, Mathematical Control Theory of Coupled PDEs seeks to provide a mathematical theory for the solution of three main problems: well-posedness and regularity of the controlled dynamics; stabilization and stability; and optimal control for both finite and infinite horizon problems along with existence/uniqueness issues of the associated Riccati equations. 1.1 Control Theory Of Dynamical Pdes 1 -- 1.1.1 Finite- Versus Infinite-dimensional Control Theory 2 -- 1.1.2 Boundary/point Control Problems For Single Pdes 2 -- 1.1.3 Boundary/point Control Problems For Systems Of Coupled Pdes 3 -- 1.2 Goal Of The Lectures 4 -- 2 Well-posedness Of Second-order Nonlinear Equations With Boundary Damping 7 -- 2.2 Abstract Model 8 -- 2.3 Existence And Uniqueness: Statement Of Main Results 9 -- 2.4 Nonlinear Plates: Von Karman Equations 13 -- 2.4.1 Case [gamma]> 0 15 -- 2.4.2 Case [gamma] = 0 19 -- 2.5 Semilinear Wave Equation 21 -- 2.6 Nonlinear Structural Acoustic Model 25 -- 2.7 Full Von Karman Systems 28 -- 2.7.1 Model 28 -- 2.7.2 Formulation Of The Results: Case [gamma] = 0 32 -- 2.7.3 Formulation Of The Results: Case [gamma]> 0 34 -- 2.8 Comments And Open Problems 37 -- 3 Uniform Stabilizability Of Nonlinear Waves And Plates 39 -- 3.2 Abstract Stabilization Inequalities 41 -- 3.3 Semilinear Wave Equation With Nonlinear Boundary Damping 45 -- 3.3.1 Formulation Of The Results 47 -- 3.3.2 Regularization 51 -- 3.3.3 Preliminary Pde Inequalities 56 -- 3.3.4 Absorption Of The Lower-order Terms 61 -- 3.3.5 Completion Of The Proof Of The Main Theorem 66 -- 3.4 Nonlinear Plate Equations 67 -- 3.4.1 Modified Von Karman Equations 67 -- 3.4.2 Full Von Karman System And Dynamic System Of Elasticity 72 -- 3.4.3 Nonlinear Plates With Thermoelasticity 77 -- 3.5 Comments And Open Problems 82 -- 4 Uniform Stability Of Structural Acoustic Models 85 -- 4.2 Internal Damping On The Wall 86 -- 4.3 Boundary Damping On The Wall 90 -- 4.3.1 Model 90 -- 4.3.2 Formulation Of The Results 92 -- 4.3.3 Preliminary Multipliers Estimates 97 -- 4.3.4 Microanalysis Estimate For The Traces Of Solutions Of Euler-bernoulli Equations And Wave Equations 101 -- 4.3.5 Observability Estimates For The Structural Acoustic Problem 104 -- 4.3.6 Completion Of The Proof Of Theorem 4.3.1 109 -- 4.4 Thermal Damping 110 -- 4.4.1 Model 110 -- 4.4.2 Statement Of Main Results 112 -- 4.4.3 Sharp Trace Regularity Results 115 -- 4.4.4 Uniform Stabilization: Proof Of Theorem 4.4.2 117 -- 4.4.5 Wave Equation 124 -- 4.4.6 Uniform Stability Analysis For The Coupled System 126 -- 4.5 Comments And Open Problems 130 -- 5 Structural Acoustic Control Problems: Semigroup And Pde Models 133 -- 5.2 Abstract Setting: Semigroup Formulation 135 -- 5.3 Pde Models Illustrating The Abstract Wall Equation (5.2.2) 140 -- 5.3.1 Plates And Beams: Flat[gamma Subscript 0] 140 -- 5.3.2 Undamped Boundary Conditions: G [identical With] 0 In (5.3.10) 143 -- 5.3.3 Boundary Feedback: Case G [not Equal] 0 In (5.3.10) And Related Stability 145 -- 5.3.4 Shells: Curved-wall [gamma Subscript 0] 151 -- 5.4 Stability In Linear Structural Acoustic Models 155 -- 5.4.1 Internal Damping On The Wall 156 -- 5.4.2 Boundary Damping On The Wall 158 -- 5.5 Comments And Open Problems 160 -- 6 Feedback Noise Control In Structural Acoustic Models: Finite Horizon Problems 163 -- 6.2 Optimal Control Problem 165 -- 6.3 Formulation Of The Results 167 -- 6.3.1 Hyperbolic-parabolic Coupling 167 -- 6.3.2 Hyperbolic-hyperbolic Coupling: General Case 168 -- 6.3.3 Hyperbolic-hyperbolic Coupling: Special Case Of The Kirchhoff Plate With Point Control 169 -- 6.4 Abstract Optimal Control Problem: General Theory 174 -- 6.4.1 Formulation Of The Abstract Control Problem 174 -- 6.4.2 Characterization Of The Optimal Control 175 -- 6.4.3 Additional Properties Under The Hyperbolic Regularity Assumption 177 -- 6.4.4 Dre, Feedback Generator, And Regularity Of The Gains B*p, B*r 179 -- 6.5 Riccati Equations Subject To The Singular Estimate For E[superscript At]b 180 -- 6.5.1 Formulation Of The Results 180 -- 6.5.2 Proof Of Lemma 6.5.1 181 -- 6.5.3 Proof Of Theorem 6.5.1 187 -- 6.6 Back To Structural Acoustic Problems: Proofs Of Theorems 6.3.1 And 6.3.2 190 -- 6.6.1 Verification Of Assumption (6.4.1) 192 -- 6.6.2 Verification Of Assumption 6.5.1 193 -- 6.7 Comments And Open Problems 201 -- 7 Feedback Noise Control In Structural Acoustic Models: Infinite Horizon Problems 203 -- 7.2 Optimal Control Problem 205 -- 7.3 Formulation Of The Results 206 -- 7.3.1 Hyperbolic-parabolic Coupling 206 -- 7.3.2 Hyperbolic-hyperbolic Coupling: Abstract Results 208 -- 7.3.3 Hyperbolic-hyperbolic Coupling: Kirchhoff Plate With Point Control 209 -- 7.4 Abstract Optimal Control Problem: General Theory 212 -- 7.4.1 Formulation Of The Abstract Control Problem 212 -- 7.4.2 Are Subject To Condition (7.4.15) 213 -- 7.5 Are Subject To A Singular Estimate For E[superscript At]b 214 -- 7.5.1 Formulation Of The Results 214 -- 7.5.2 Proof Of Theorem 7.5.1 215 -- 7.6 Back To Structural Acoustic Problems: Proofs Of Theorems 7.3.1 And 7.3.2 222. Irena Lasiecka. Includes Bibliographical References And Index.