Optimal Control of Systems Governed by Partial Differential Equations

1. The development of a theory of optimal control (deterministic) requires the following initial data: (i) a control u belonging to some set ilIi ad (the set of 'admissible controls') which is at our disposition, (ii) for a given control u, the state y(u) of the system which is to be controlled is given by the solution of an equation (*) Ay(u)=given function ofu where A is an operator (assumed known) which specifies the system to be controlled (A is the 'model' of the system), (iii) the observation z(u) which is a function of y(u) (assumed to be known exactly; we consider only deterministic problems in this book), (iv) the "cost function" J(u) ("economic function") which is defined in terms of a numerical function z-+

Principal Notations.- I Minimization of Functions and Unilateral Boundary Value Problems.- 1. Minimization of Coercive Forms.- 2. A Direct Solution of Certain Variational Inequalities.- 3. Examples.- 4. A Comparison Theorem.- 5. Non Coercive Forms.- Notes.- II Control of Systems Governed by Elliptic Partial Differential Equations.- 1. Control of Elliptic Variational Problems.- 2. First Applications.- 3. A Family of Examples with N = 0 and $$ {U_{ad}} $$ Arbitrary.- 4. Observation on the Boundary.- 5. Control and Observation on the Boundary. Case of the Dirichlet Problem.- 6. Constraints on the State.- 7. Existence Results for Optimal Controls.- 8. First Order Necessary Conditions.- Notes.- III Control of Systems Governed by Parabolic Partial Differential Equations.- 1. Equations of Evolution.- 2. Problems of Control.- 3. Examples.- 4. Decoupling and Integro-Differential Equation of Riccati Type (I).- 5. Decoupling and Integro-Differential Equation of Riccati Type (II).- 6. Behaviour asT ? + ?.- 7. Problems which are not Necessarily Coercive.- 8. Other Observations of the State and other Types of Control.- 9. Boundary Control and Observation on the Boundary or of the Final State for a System Governed by a Mixed Dirichlet Problem.- 10. Controllability.- 11. Control via Initial Conditions; Estimation.- 12. Duality.- 13. Constraints on the Control and the State.- 14. Non Quadratic Cost Functions.- 15. Existence Results for Optimal Controls.- 16. First Order Necessary Conditions.- 17. Time Optimal Control.- 18. Miscellaneous.- 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.- 2. Control Problems.- 3. Transposition and Applications to Control.- 4. Examples.- 5. Decoupling.- 6. Control via Initial Conditions. Estimation.- 7. Boundary Control (I).- 8. Boundary Control (II).- 9. Parabolic-Hyperbolic Systems.- 10. Existence Theorems.- Notes.- V Regularization, Approximation and Penalization.- 1. Regularization.- 2. Approximation in Terms of Systems of Cauchy-Kowaleska Type.- 3. Penalization.- Notes.