Optimal Control of Systems Governed by Partial Differential Equations.

Abstract

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 $$ {U_{ad}} $$.- 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 $$ {U_{ad}} $$ 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 th