Optimal control of systems governed by partial differential equations

著者

• Lions, Jacques Louis
• Mitter, S. K. (Sanjoy K.)

書誌事項

Optimal control of systems governed by partial differential equations

J.L. Lions ; translated by S.K. Mitter

（Die Grundlehren der mathematischen Wissenschaften, Bd. 170）

Springer-Verlag, 1971

• : gw
• : us
• : softcover

タイトル別名

Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles

Optimal control of systems

注記

Originally published as Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles: Paris : Dunod : Gauthier-Villars, 1968. (Études mathémathiques)

Bibliography: p. [384]-394

Includes index

内容説明・目次

巻冊次

: gw ISBN 9783540051152

内容説明

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-+

目次

• 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 Equation.- 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 Riccati 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 Riccati 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 Riccati 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 $$ {U_{ad}} $$ 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 $$ \frac{<!-- -->{\partial y}}{<!-- -->{\partial {v_A}}} $$
• 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. Riccati Integro-differential Equation.- 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 Equation.- 3. Penalization.- Notes.

巻冊次

: softcover ISBN 9783642650260

内容説明

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-+

目次

• 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 Equation.- 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 Riccati 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 Riccati 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 Riccati 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 $$ {U_{ad}} $$ 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 $$ \frac{<!-- -->{\partial y}}{<!-- -->{\partial {v_A}}} $$
• 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. Riccati Integro-differential Equation.- 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 Equation.- 3. Penalization.- Notes.

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詳細情報

• NII書誌ID(NCID): BA03958952
• ISBN:
  • 3540051155
  • 0387051155
  • 9783642650260
• LCCN: 78113638
• 出版国コード: gw
• タイトル言語コード: eng
• 本文言語コード: eng
• 原本言語コード: fre
• 出版地: Berlin ; New York
• ページ数/冊数: xi, 396 p.
• 大きさ: 24 cm
• 分類:
  • LCC : QA402.3
  • DC : 629.8/312
• 件名:
  • LCSH : Control theory
  • LCSH : Differential equations, Partial
• 親書誌ID:
  • BA00007422