Optimal Control of Systems Governed by Partial Differential Equations de Jacques Louis Lions

Optimal Control of Systems Governed by Partial Differential Equations

Jacques Louis Lions

Sanjog K. Mitter

en Limba Engleză Paperback – 12 noi 2011

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-+

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Cuprins

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.