Optimal Control of Systems Governed by Partial Differential Equations.

Principal Notations. - I Minimization of Functions and Unilateral Boundary Value Problems. - 1. Minimization of Coercive Forms. - 1. 1. Notation. - 1. 2. The Case when? : is Coercive. - 1. 3. Characterization of the Minimizing Element. Variational Inequalities. - 1. 4. Alternative Form of Variational Inequalities. - 1. 5. Function J being the Sum of a Differentiable and Non-Differentiable Function. - 1. 6. The Convexity Hypothesis on U₀₃. - 1. 7. Orientation. - 2. A Direct Solution of Certain Variational Inequalities. - 2. 1. Problem Statement. - 2. 2. An Existence and Uniqueness Theorem. - 3. Examples. - 3. 1. Function Spaces on? . - 3. 2. Function Spaces on? . - 3. 3. Subspaces of Hm (? ). - 3. 4. Examples of Boundary Value Problems. - 3. 5. Unilateral Boundary Value Problems (I). - 3. 6. Unilateral Boundary Value Problems (II). - 3. 7. Unilateral Boundary Value Problems (III). - 3. 8. Unilateral Boundary Value Problems Case of Systems. - 3. 9. Elliptic Operators of Order Greater than Two. - 3. 10. Non-differentiable Functionals. - 4. A Comparison Theorem. - 4. 1. General Results. - 4. 2. An Application. - 5. Non Coercive Forms. - 5. 1. Convexity of the Set of Solutions. - 5. 2. Approximation Theorem. - Notes. - II Control of Systems Governed by Elliptic Partial Differential Equations. - 1. Control of Elliptic Variational Problems. - 1. 1. Problem Statement. - 1. 2. First Remarks on the Control Problem. - 1. 3. The Set of Inequalities Defining the Optimal Control. - 2. First Applications. - 2. 1. System Governed by the Dirichlet Problem Distributed Control. - 2. 2. The Case with No Constraints. - 2. 3. System Governed by a Neumann Problem Distributed Control. - 2. 4. System Governed by a Neumann Problem Boundary Control. - 2. 5. Local and Global Constraints. - 2. 6. System Governed by a Differential System. - 2. 7. System Governed by a 4th Order Differential Operator. - 2. 8. Orientation. - 3. A Family of Examples with N = 0 and U₀₃ Arbitrary. - 3. 1. General Case. - 3. 2. Application (I). - 3. 3. Application (II). - 4. Observation on the Boundary. - 4. 1. System Governed by a Dirichlet Problem (I). - 4. 2. Some Results on Non-homogeneous Dirichlet Problems. - 4. 3. System Governed by a Dirichlet Problem (II). - 4. 4. System Governed by a Neumann Problem. - 5. Control and Observation on the Boundary. Case of the Dirichlet Problem. - 5. 1. Orientation. - 5. 2. Boundary Control in L2 (? ). - 5. 3. A Controllability-Like Problem. - 5. 4. Pointwise Control and Observation. - 6. Constraints on the State. - 6. 1. Orientation. - 6. 2. Control and Constraints on the Boundary. - 7. Existence Results for Optimal Controls. - 7. 1. Orientation. - 7. 2. Distributed Control. - 7. 3. Singular Perturbation of the System. - 7. 4. Boundary Control. - 7. 5. Control of Systems Governed by Unilateral Problems. - 8. First Order Necessary Conditions. - 8. 1. Statement of the Theorem. - 8. 2. Proof of the Theorem. - 8. 2. 1. Algebraic Transformation. - 8. 2. 2. General Remarks on the Utilization of (8. 13. ). - 8. 2. 3. Proof that dj,? 0. - Notes. - III Control of Systems Governed by Parabolic Partial Differential Equations. - 1. Equations of Evolution. - 1. 1. Data. - 1. 2. Evolution Problems. - 1. 3. Proof of Uniqueness. - 1. 4. Proof of Existence. - 1. 5. Some Examples. - 1. 6. Semi-groups. - 2. Problems of Control. - 2. 1. Notation. Immediate Properties. - 2. 2. Set of Inequalities Characterizing the Optimal Control. - 2. 3. Case (i). Set of Inequalities. - 2. 4. Case (ii). Set of Inequalities. - 2. 5. Orientation. - 3. Examples. - 3. 1. Mixed Dirichlet Problem for a Second Order Parabolic 3. 1. 1. C = Injection Map of L2 (0, T V)? L2 (Q). - 3. 1. 2. C = Identity Map of L2 (0, T V) into itself. - 3. 1. 3. Observation of the Final State. - 3. 2. Mixed Neumann Problem for a Parabolic Equation of Second Order. - 3. 2. 1. Case (i). - 3. 2. 2. Case (ii). - 3. 3. System of Equations and Equations of Higher Order. - 3. 3. 1. System of Equations. - 3. 3. 2. Higher Order Equations. - 3. 4. Additional Results. - 3. 5. Orientation. - 4. Decoupling and Integro-Differential Equation of Type (I). - 4. 1. Notation and Assumptions. - 4. 2. Operator P (t), Function r (t). - 4. 3. Formal Calculations. - 4. 4. The Finite Dimensional Case Approximation. - 4. 5. Passage to the Limit. - 4. 6. Integro-Differential Equation of Type. - 4. 7. Connections with the Hamilton-Jacobi Theory. - 4. 8. The Case where Constraints are Present. - 4. 9. Various Remarks. - 4. 9. 1. Direct Study of the Riccati Equation. - 4. 9. 2. Another Approach to the Direct Study of the Riccati Equation. - 4. 9. 3. Yet Another Approach to the Direct Study of the Riccati Equation. - 5. Decoupling and Integro-Differential Equation of Type (II). - 5. 1. Application of the Schwartz-Kernel Theorem. - 5. 2. Example of a Mixed Neumann Problem with Boundary Control. - 5. 3. Example of a Mixed Neumann Problem with Observation of the Final State. - 5. 4. Mixed Neumann Problem, Observation of the Final State and Constraints in a Vector Space. - 5. 5. Remarks on Decoupling in the Presence of Constraints. - 6. Behaviour as T? +? . - 6. 1. Orientation and Hypotheses. - 6. 2. The Case T =? . - 6. 3. Passage to the Limit as T? +? . - 7. Problems which are not Necessarily Coercive. - 7. 1. Distributed Observation. - 7. 2. Observation of the Final State. - 7. 3. Examples where N = 0 and U₀₃ is not Bounded. - 8. Other Observations of the State and other Types of Control. - 8. 1. Pointwise Observation of the State. - 8. 2. Pointwise Control. - 8. 3. Control and Observation on the Boundary. - 9. Boundary Control and Observation on the Boundary or of the Final State for a System Governed by a Mixed Dirichlet Problem. - 9. 1. Orientation and Problem Statement. - 9. 2. Non Homogeneous Mixed Dirichlet Problem. - 9. 3. Definition of y{ {vA}} Observation. - 9. 4. Cost Function Equations of Optimal Control. - 9. 5. Regular Control. - 9. 6. Observation of the Final State. - 9. 7. Observation of the Final State, Second Order Parabolic Operator. - 10. Controllability. - 10. 1. Problem Statement. - 10. 2. Controllability and Uniqueness. - 10. 3. Super-Controllability and Super-Uniqueness. - 11. Control via Initial Conditions Estimation. - 11. 1. Problem Statement. General Results. - 11. 2. Examples. - 11. 3. Controllability. - 11. 4. An Estimation Problem. - 12. Duality. - 12. 1. General Remarks. - 12. 2. Example. - 13. Constraints on the Control and the State. - 13. 1. A General Result. - 13. 2. Applications (I). - 13. 3. Applications (II). - 14. Non Quadratic Cost Functions. - 14. 1. Orientation. - 14. 2. An Example. - 14. 3. Remarks on Decoupling. - 15. Existence Results for Optimal Controls. - 15. 1. Orientation. - 15. 2. Non-linear Problem with Distributed Control (I). - 15. 3. Non-linear Problem with Distributed Control. Singular Perturbation. - 15. 4. Non-linear Problem. Boundary Control. - 15. 5. Utilization of Convexity and the Maximum Principle for Second Order Parabolic Equations. - 15. 6. Control of Systems Governed by Evolution Inequalities. - 16. First Order Necessary Conditions. - 16. 1. Statement of the Theorem. - 16. 2. Proof of Theorem 16. 1. - 16. 2. 1. Algebraic Transformation. - 16. 2. 2. Utilization of (16. 11. ). - 16. 2. 3. Proof of (16. 12. ). - 16. 3. Remarks. - 17. Time Optimal Control. - 17. 1. Problem Statement. - 17. 2. Existence Theorem. - 17. 3. Bang-Bang Theorem. - 18. Miscellaneous. - 18. 1. Equations with Delay. - 18. 1. 1. Definition of the State. - 18. 1. 2. Control Problem. - 18. 2. Spaces which are not Normable. - Notes. - IV Control of Systems Governed by Hyperbolic Equations or by Equations which are well Posed in the Petrowsky Sense. - 1. Second Order Evolution Equations. - 1. 1. Notation and Hypotheses. - 1. 2. Problem Statement. An Existence and Uniqueness Result. - 1. 3. Proof of Uniqueness. - 1. 4. Proof of Existence. - 1. 5. Examples (I). - 1. 6. Examples (II). - 1. 7. Orientation. - 2. Control Problems. - 2. 1. Notation. Immediate Properties. - 2. 2. Case (2. 5. ). - 2. 3. Case (2. 6. ). - 2. 4. Case (2. 7. ). - 2. 5. Case (2. 8. ). - 3. Transposition and Applications to Control. - 3. 1. Transposition of Theorem 1. 1. - 3. 2. Application (I). - 3. 3. Application (II). - 3. 4. Application (III). - 4. Examples. - 4. 1. Examples of Hyperbolic Problems. Distributed Control, Distributed Observation. - 4. 2. Examples of Hyperbolic Systems. Distributed Control, Observation of the Final State. - 4. 3. Petrowsky Type Equation. Distributed Control. Distributed Observation. - 4. 4. Petrowsky Type Equation. Distributed Control. Observation of the Final State. - 4. 5. Orientation. - 5. Decoupling. - 5. 1. Problem Statement. Rewriting as a System of First Order Equations. - 5. 2. Rewriting of the Set of Equations Determining the Optimal Control. - 5. 3. Decoupling. - 5. 4. Integro-differential 5. 5. Another Optimal Control Problem. Decoupling. - 6. Control via Initial Conditions. Estimation. - 6. 1. Problem Statement. - 6. 2. Coercivity of J (? ). - 6. 3. System of Equations Determining the Optimal Control. - 7. Boundary Control (I). - 7. 1. Problem Statement. - 7. 2. Definition of the State of the System. - 7. 3. Distributed Observation. - 7. 4. Boundary Observation. - 8. Boundary Control (II). - 8. 1. Problem Statement. - 8. 2. Control? Regular. - 8. 3. Examples. - 9. Parabolic-Hyperbolic Systems. - 9. 1. Recapitulation of Some General Results. - 9. 2. Complement. - 9. 3. Control Problems. - 9. 4. Example (I). - 9. 5. Example (II). - 9. 6. Decoupling. - 10. Existence Theorems. - 10. 1. Orientation. - 10. 2. Example. Introduction of a Viscosity Term. - 10. 3. Time Optimal Control. - Notes. - V Regularization, Approximation and Penalization. - 1. Regularization. - 1. 1. Parabolic Regularization. - 1. 2. Application to Optimal Control. - 1. 3. Application to Decoupling. - 1. 4. Various Remarks. - 1. 5. Regularization of the Control. - 2. Approximation in Terms of Systems of Cauchy-Kowaleska Type. - 2. 1. Evolution Equation on a Variety. - 2. 2. Approximation by a System of Cauchy-Kowaleska Type. - 2. 3. Linearized Navier-Stokes 3. Penalization. - Notes.