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Foreword; Preface; Acknowledgments; Contents; 1 Introduction; 1.1 Controlling PDEs: Why Point Controls and Sensors?; 1.2 Observability and Controllability: Methodology of Duality; 1.2.1 Controllability Problem; 1.2.2 Observability Problem; 1.3 Degenerate Sensors and Controls: Challenges and Proposed Methodology to Address Them; 1.3.1 Observability Problem with Static Point Sensors: Principal Difficulties; Part I Observability and Controllability of Linear Parabolic Equations by Means of Degenerate Sensors and Controls
3.5 Observability and ControllabilityPart II Observability and Controllability of Semilinear Parabolic Equations by Means of Degenerate Sensors and Controls; 4 Behavior of Solutions of the Semilinear Heat Equation in Vanishing Time and Controllability; 4.1 Introduction; 4.2 Auxiliary A Priori Estimates; 4.3 Approximate Controllability Results; 4.4 A Priori Estimates for u L∞(QT) and Further Results; 4.5 Examples; 4.5.1 Discussion of Examples; Appendix: Proof of Lemma 4.4; 5 Controllability of the Semilinear Heat Equation with a Sublinear Term and a Degenerate Actuator; 5.1 Introduction
5.2 Proof of Theorem 5.15.3 Proof of Theorem 5.2; 5.4 The Case of Several Spatial Dimensions; 6 Controllability of the Semilinear Reaction-Diffusion Equation with a Degenerate Actuator; 6.1 Introduction; 6.2 Main Results; 6.2.1 The Case of f = f(u); 6.2.2 The Case of f= f(x,t, u); 6.2.3 Controllability in C0 [0, 1] and L∞(0,1); 6.2.4 An Example of Superlinear f; 6.3 Some Auxiliary Formulas; 6.4 Proofs of Theorems 6.1 and 6.2; 6.5 Proof of Theorem 2.3; 6.6 Proof of Theorem 6.5; 7 Semilinear Parabolic Equations: Mobile Point Controls Versus Locally Distributed Ones; 7.1 Introduction
7.2 The Linear Case: Duality7.3 Continuous Observability Estimate for Mobile Point Sensors; 7.4 Continuous Observability Estimate for Locally Distributed Sensors; 7.5 The Semilinear Case; 7.6 Examples; 7.7 Possible Extensions of the Main Results; Appendix: Proof of Theorem 7.4; Part III Source Localization and Sensor Placement in Environmental Monitoring; 8 Degenerate Sensors in Source Localization and Sensor Placement Problems; 8.1 Introduction; 8.2 A Model with Single-Point Source and Single-Point Sensor; 8.3 1-D Reaction-Diffusion-Convection Model
3.5 Observability and ControllabilityPart II Observability and Controllability of Semilinear Parabolic Equations by Means of Degenerate Sensors and Controls; 4 Behavior of Solutions of the Semilinear Heat Equation in Vanishing Time and Controllability; 4.1 Introduction; 4.2 Auxiliary A Priori Estimates; 4.3 Approximate Controllability Results; 4.4 A Priori Estimates for u L∞(QT) and Further Results; 4.5 Examples; 4.5.1 Discussion of Examples; Appendix: Proof of Lemma 4.4; 5 Controllability of the Semilinear Heat Equation with a Sublinear Term and a Degenerate Actuator; 5.1 Introduction
5.2 Proof of Theorem 5.15.3 Proof of Theorem 5.2; 5.4 The Case of Several Spatial Dimensions; 6 Controllability of the Semilinear Reaction-Diffusion Equation with a Degenerate Actuator; 6.1 Introduction; 6.2 Main Results; 6.2.1 The Case of f = f(u); 6.2.2 The Case of f= f(x,t, u); 6.2.3 Controllability in C0 [0, 1] and L∞(0,1); 6.2.4 An Example of Superlinear f; 6.3 Some Auxiliary Formulas; 6.4 Proofs of Theorems 6.1 and 6.2; 6.5 Proof of Theorem 2.3; 6.6 Proof of Theorem 6.5; 7 Semilinear Parabolic Equations: Mobile Point Controls Versus Locally Distributed Ones; 7.1 Introduction
7.2 The Linear Case: Duality7.3 Continuous Observability Estimate for Mobile Point Sensors; 7.4 Continuous Observability Estimate for Locally Distributed Sensors; 7.5 The Semilinear Case; 7.6 Examples; 7.7 Possible Extensions of the Main Results; Appendix: Proof of Theorem 7.4; Part III Source Localization and Sensor Placement in Environmental Monitoring; 8 Degenerate Sensors in Source Localization and Sensor Placement Problems; 8.1 Introduction; 8.2 A Model with Single-Point Source and Single-Point Sensor; 8.3 1-D Reaction-Diffusion-Convection Model