Optimal Control of ODEs and DAEs.:
The intention of this textbook is to provide both, the theoretical and computational tools that are necessary to investigate and to solve optimal control problems with ordinary differential equations and differential-algebraic equations. An emphasis is placed on the interplay between the continuous...
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| 1. Verfasser: | |
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Berlin :
De Gruyter,
2012.
|
| Schriftenreihe: | De Gruyter textbook.
|
| Schlagworte: | |
| Online-Zugang: | Volltext Volltext |
| Zusammenfassung: | The intention of this textbook is to provide both, the theoretical and computational tools that are necessary to investigate and to solve optimal control problems with ordinary differential equations and differential-algebraic equations. An emphasis is placed on the interplay between the continuous optimal control problem, which typically is defined and analyzed in a Banach space setting, and discrete optimal control problems, which are obtained by discretization and lead to finite dimensional optimization problems. |
| Beschreibung: | 5.3.2 Adjoint Equation Approach: The Discrete Case. |
| Beschreibung: | 1 online resource (468 pages) |
| Bibliographie: | Includes bibliographical references and index. |
| ISBN: | 9783110249996 3110249995 |
Internformat
MARC
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| 100 | 1 | |a Gerdts, Matthias. | |
| 245 | 1 | 0 | |a Optimal Control of ODEs and DAEs. |
| 260 | |a Berlin : |b De Gruyter, |c 2012. | ||
| 300 | |a 1 online resource (468 pages) | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 490 | 1 | |a De Gruyter textbook | |
| 505 | 0 | |a Preface; 1 Introduction; 1.1 DAE Optimal Control Problems; 1.1.1 Perturbation Index; 1.1.2 Consistent Initial Values; 1.1.3 Index Reduction and Stabilization; 1.2 Transformation Techniques; 1.2.1 Transformation to Fixed Time Interval; 1.2.2 Transformation to Autonomous Problem; 1.2.3 Transformation of Tschebyscheff Problems; 1.2.4 Transformation of L 1 -Minimization Problems; 1.2.5 Transformation of Interior-Point Constraints; 1.3 Overview; 1.4 Exercises; 2 Infinite Optimization Problems; 2.1 Function Spaces; 2.1.1 Topological Spaces, Banach Spaces, and Hilbert Spaces. | |
| 505 | 8 | |a 2.1.2 Mappings and Dual Spaces2.1.3 Derivatives, Mean-Value Theorem, and Implicit Function Theorem; 2.1.4 Lp-Spaces, Wq; P-Spaces, Absolutely Continuous Functions, Functions of Bounded Variation; 2.2 The DAE Optimal Control Problem as an Infinite Optimization Problem; 2.3 Necessary Conditions for Infinite Optimization Problems; 2.3.1 Existence of a Solution; 2.3.2 Conic Approximation of Sets; 2.3.3 Separation Theorems; 2.3.4 First Order Necessary Optimality Conditions of Fritz John Type; 2.3.5 Constraint Qualifications and Karush-Kuhn-Tucker Conditions; 2.4 Exercises. | |
| 505 | 8 | |a 3 Local Minimum Principles3.1 Problems without Pure State and Mixed Control-State Constraints; 3.1.1 Representation of Multipliers; 3.1.2 Local Minimum Principle; 3.1.3 Constraint Qualifications and Regularity; 3.2 Problems with Pure State Constraints; 3.2.1 Representation of Multipliers; 3.2.2 Local Minimum Principle; 3.2.3 Finding Controls on Active State Constraint Arcs; 3.2.4 Jump Conditions for the Adjoint; 3.3 Problems with Mixed Control-State Constraints; 3.3.1 Representation of Multipliers; 3.3.2 Local Minimum Principle; 3.4 Summary of Local Minimum Principles for Index-One Problems. | |
| 505 | 8 | |a 3.5 Exercises4 Discretization Methods for ODEs and DAEs; 4.1 Discretization by One-Step Methods; 4.1.1 The Euler Method; 4.1.2 Runge-Kutta Methods; 4.1.3 General One-Step Method; 4.1.4 Consistency, Stability, and Convergence of One-Step Methods; 4.2 Backward Differentiation Formulas (BDF); 4.3 Linearized Implicit Runge-Kutta Methods; 4.4 Automatic Step-size Selection; 4.5 Computation of Consistent Initial Values; 4.5.1 Projection Method for Consistent Initial Values; 4.5.2 Consistent Initial Values via Relaxation; 4.6 Shooting Techniques for Boundary Value Problems. | |
| 505 | 8 | |a 4.6.1 Single Shooting Method using Projections4.6.2 Single Shooting Method using Relaxations; 4.6.3 Multiple Shooting Method; 4.7 Exercises; 5 Discretization of Optimal Control Problems; 5.1 Direct Discretization Methods; 5.1.1 Full Discretization Approach; 5.1.2 Reduced Discretization Approach; 5.1.3 Control Discretization; 5.2 A Brief Introduction to Sequential Quadratic Programming; 5.2.1 Lagrange-Newton Method; 5.2.2 Sequential Quadratic Programming (SQP); 5.3 Calculation of Derivatives for Reduced Discretization; 5.3.1 Sensitivity Equation Approach. | |
| 500 | |a 5.3.2 Adjoint Equation Approach: The Discrete Case. | ||
| 520 | |a The intention of this textbook is to provide both, the theoretical and computational tools that are necessary to investigate and to solve optimal control problems with ordinary differential equations and differential-algebraic equations. An emphasis is placed on the interplay between the continuous optimal control problem, which typically is defined and analyzed in a Banach space setting, and discrete optimal control problems, which are obtained by discretization and lead to finite dimensional optimization problems. | ||
| 588 | 0 | |a Print version record. | |
| 504 | |a Includes bibliographical references and index. | ||
| 546 | |a English. | ||
| 650 | 0 | |a Control theory |x Mathematical models. | |
| 650 | 0 | |a Mathematical optimization. |0 http://id.loc.gov/authorities/subjects/sh85082127 | |
| 650 | 6 | |a Théorie de la commande |x Modèles mathématiques. | |
| 650 | 6 | |a Optimisation mathématique. | |
| 650 | 7 | |a MATHEMATICS |x Calculus. |2 bisacsh | |
| 650 | 7 | |a MATHEMATICS |x Mathematical Analysis. |2 bisacsh | |
| 650 | 7 | |a Control theory |x Mathematical models |2 fast | |
| 650 | 7 | |a Mathematical optimization |2 fast | |
| 758 | |i has work: |a Optimal control of ODEs and DAEs (Text) |1 https://id.oclc.org/worldcat/entity/E39PCFvppCbFCwwkgRrWHhTT73 |4 https://id.oclc.org/worldcat/ontology/hasWork | ||
| 776 | 0 | 8 | |i Print version: |a Gerdts, Matthias. |t Optimal Control of ODEs and DAEs. |d Berlin : De Gruyter, ©2012 |z 9783110249958 |
| 830 | 0 | |a De Gruyter textbook. | |
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Datensatz im Suchindex
| DE-BY-FWS_katkey | ZDB-4-EBU-ocn772845160 |
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| _version_ | 1816796903198687232 |
| adam_text | |
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| author | Gerdts, Matthias |
| author_facet | Gerdts, Matthias |
| author_role | |
| author_sort | Gerdts, Matthias |
| author_variant | m g mg |
| building | Verbundindex |
| bvnumber | localFWS |
| callnumber-first | Q - Science |
| callnumber-label | QA402 |
| callnumber-raw | QA402.3 .G444 2012 |
| callnumber-search | QA402.3 .G444 2012 |
| callnumber-sort | QA 3402.3 G444 42012 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 880 |
| collection | ZDB-4-EBU |
| contents | Preface; 1 Introduction; 1.1 DAE Optimal Control Problems; 1.1.1 Perturbation Index; 1.1.2 Consistent Initial Values; 1.1.3 Index Reduction and Stabilization; 1.2 Transformation Techniques; 1.2.1 Transformation to Fixed Time Interval; 1.2.2 Transformation to Autonomous Problem; 1.2.3 Transformation of Tschebyscheff Problems; 1.2.4 Transformation of L 1 -Minimization Problems; 1.2.5 Transformation of Interior-Point Constraints; 1.3 Overview; 1.4 Exercises; 2 Infinite Optimization Problems; 2.1 Function Spaces; 2.1.1 Topological Spaces, Banach Spaces, and Hilbert Spaces. 2.1.2 Mappings and Dual Spaces2.1.3 Derivatives, Mean-Value Theorem, and Implicit Function Theorem; 2.1.4 Lp-Spaces, Wq; P-Spaces, Absolutely Continuous Functions, Functions of Bounded Variation; 2.2 The DAE Optimal Control Problem as an Infinite Optimization Problem; 2.3 Necessary Conditions for Infinite Optimization Problems; 2.3.1 Existence of a Solution; 2.3.2 Conic Approximation of Sets; 2.3.3 Separation Theorems; 2.3.4 First Order Necessary Optimality Conditions of Fritz John Type; 2.3.5 Constraint Qualifications and Karush-Kuhn-Tucker Conditions; 2.4 Exercises. 3 Local Minimum Principles3.1 Problems without Pure State and Mixed Control-State Constraints; 3.1.1 Representation of Multipliers; 3.1.2 Local Minimum Principle; 3.1.3 Constraint Qualifications and Regularity; 3.2 Problems with Pure State Constraints; 3.2.1 Representation of Multipliers; 3.2.2 Local Minimum Principle; 3.2.3 Finding Controls on Active State Constraint Arcs; 3.2.4 Jump Conditions for the Adjoint; 3.3 Problems with Mixed Control-State Constraints; 3.3.1 Representation of Multipliers; 3.3.2 Local Minimum Principle; 3.4 Summary of Local Minimum Principles for Index-One Problems. 3.5 Exercises4 Discretization Methods for ODEs and DAEs; 4.1 Discretization by One-Step Methods; 4.1.1 The Euler Method; 4.1.2 Runge-Kutta Methods; 4.1.3 General One-Step Method; 4.1.4 Consistency, Stability, and Convergence of One-Step Methods; 4.2 Backward Differentiation Formulas (BDF); 4.3 Linearized Implicit Runge-Kutta Methods; 4.4 Automatic Step-size Selection; 4.5 Computation of Consistent Initial Values; 4.5.1 Projection Method for Consistent Initial Values; 4.5.2 Consistent Initial Values via Relaxation; 4.6 Shooting Techniques for Boundary Value Problems. 4.6.1 Single Shooting Method using Projections4.6.2 Single Shooting Method using Relaxations; 4.6.3 Multiple Shooting Method; 4.7 Exercises; 5 Discretization of Optimal Control Problems; 5.1 Direct Discretization Methods; 5.1.1 Full Discretization Approach; 5.1.2 Reduced Discretization Approach; 5.1.3 Control Discretization; 5.2 A Brief Introduction to Sequential Quadratic Programming; 5.2.1 Lagrange-Newton Method; 5.2.2 Sequential Quadratic Programming (SQP); 5.3 Calculation of Derivatives for Reduced Discretization; 5.3.1 Sensitivity Equation Approach. |
| ctrlnum | (OCoLC)772845160 |
| dewey-full | 515.642 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.642 |
| dewey-search | 515.642 |
| dewey-sort | 3515.642 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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1 Introduction; 1.1 DAE Optimal Control Problems; 1.1.1 Perturbation Index; 1.1.2 Consistent Initial Values; 1.1.3 Index Reduction and Stabilization; 1.2 Transformation Techniques; 1.2.1 Transformation to Fixed Time Interval; 1.2.2 Transformation to Autonomous Problem; 1.2.3 Transformation of Tschebyscheff Problems; 1.2.4 Transformation of L 1 -Minimization Problems; 1.2.5 Transformation of Interior-Point Constraints; 1.3 Overview; 1.4 Exercises; 2 Infinite Optimization Problems; 2.1 Function Spaces; 2.1.1 Topological Spaces, Banach Spaces, and Hilbert Spaces.</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">2.1.2 Mappings and Dual Spaces2.1.3 Derivatives, Mean-Value Theorem, and Implicit Function Theorem; 2.1.4 Lp-Spaces, Wq; P-Spaces, Absolutely Continuous Functions, Functions of Bounded Variation; 2.2 The DAE Optimal Control Problem as an Infinite Optimization Problem; 2.3 Necessary Conditions for Infinite Optimization Problems; 2.3.1 Existence of a Solution; 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4.1 Discretization by One-Step Methods; 4.1.1 The Euler Method; 4.1.2 Runge-Kutta Methods; 4.1.3 General One-Step Method; 4.1.4 Consistency, Stability, and Convergence of One-Step Methods; 4.2 Backward Differentiation Formulas (BDF); 4.3 Linearized Implicit Runge-Kutta Methods; 4.4 Automatic Step-size Selection; 4.5 Computation of Consistent Initial Values; 4.5.1 Projection Method for Consistent Initial Values; 4.5.2 Consistent Initial Values via Relaxation; 4.6 Shooting Techniques for Boundary Value Problems.</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">4.6.1 Single Shooting Method using Projections4.6.2 Single Shooting Method using Relaxations; 4.6.3 Multiple Shooting Method; 4.7 Exercises; 5 Discretization of Optimal Control Problems; 5.1 Direct Discretization Methods; 5.1.1 Full Discretization Approach; 5.1.2 Reduced Discretization Approach; 5.1.3 Control Discretization; 5.2 A Brief Introduction to Sequential Quadratic Programming; 5.2.1 Lagrange-Newton Method; 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| id | ZDB-4-EBU-ocn772845160 |
| illustrated | Not Illustrated |
| indexdate | 2024-11-26T14:49:04Z |
| institution | BVB |
| isbn | 9783110249996 3110249995 |
| language | English |
| oclc_num | 772845160 |
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| owner_facet | MAIN DE-863 DE-BY-FWS |
| physical | 1 online resource (468 pages) |
| psigel | ZDB-4-EBU |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | De Gruyter, |
| record_format | marc |
| series | De Gruyter textbook. |
| series2 | De Gruyter textbook |
| spelling | Gerdts, Matthias. Optimal Control of ODEs and DAEs. Berlin : De Gruyter, 2012. 1 online resource (468 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier De Gruyter textbook Preface; 1 Introduction; 1.1 DAE Optimal Control Problems; 1.1.1 Perturbation Index; 1.1.2 Consistent Initial Values; 1.1.3 Index Reduction and Stabilization; 1.2 Transformation Techniques; 1.2.1 Transformation to Fixed Time Interval; 1.2.2 Transformation to Autonomous Problem; 1.2.3 Transformation of Tschebyscheff Problems; 1.2.4 Transformation of L 1 -Minimization Problems; 1.2.5 Transformation of Interior-Point Constraints; 1.3 Overview; 1.4 Exercises; 2 Infinite Optimization Problems; 2.1 Function Spaces; 2.1.1 Topological Spaces, Banach Spaces, and Hilbert Spaces. 2.1.2 Mappings and Dual Spaces2.1.3 Derivatives, Mean-Value Theorem, and Implicit Function Theorem; 2.1.4 Lp-Spaces, Wq; P-Spaces, Absolutely Continuous Functions, Functions of Bounded Variation; 2.2 The DAE Optimal Control Problem as an Infinite Optimization Problem; 2.3 Necessary Conditions for Infinite Optimization Problems; 2.3.1 Existence of a Solution; 2.3.2 Conic Approximation of Sets; 2.3.3 Separation Theorems; 2.3.4 First Order Necessary Optimality Conditions of Fritz John Type; 2.3.5 Constraint Qualifications and Karush-Kuhn-Tucker Conditions; 2.4 Exercises. 3 Local Minimum Principles3.1 Problems without Pure State and Mixed Control-State Constraints; 3.1.1 Representation of Multipliers; 3.1.2 Local Minimum Principle; 3.1.3 Constraint Qualifications and Regularity; 3.2 Problems with Pure State Constraints; 3.2.1 Representation of Multipliers; 3.2.2 Local Minimum Principle; 3.2.3 Finding Controls on Active State Constraint Arcs; 3.2.4 Jump Conditions for the Adjoint; 3.3 Problems with Mixed Control-State Constraints; 3.3.1 Representation of Multipliers; 3.3.2 Local Minimum Principle; 3.4 Summary of Local Minimum Principles for Index-One Problems. 3.5 Exercises4 Discretization Methods for ODEs and DAEs; 4.1 Discretization by One-Step Methods; 4.1.1 The Euler Method; 4.1.2 Runge-Kutta Methods; 4.1.3 General One-Step Method; 4.1.4 Consistency, Stability, and Convergence of One-Step Methods; 4.2 Backward Differentiation Formulas (BDF); 4.3 Linearized Implicit Runge-Kutta Methods; 4.4 Automatic Step-size Selection; 4.5 Computation of Consistent Initial Values; 4.5.1 Projection Method for Consistent Initial Values; 4.5.2 Consistent Initial Values via Relaxation; 4.6 Shooting Techniques for Boundary Value Problems. 4.6.1 Single Shooting Method using Projections4.6.2 Single Shooting Method using Relaxations; 4.6.3 Multiple Shooting Method; 4.7 Exercises; 5 Discretization of Optimal Control Problems; 5.1 Direct Discretization Methods; 5.1.1 Full Discretization Approach; 5.1.2 Reduced Discretization Approach; 5.1.3 Control Discretization; 5.2 A Brief Introduction to Sequential Quadratic Programming; 5.2.1 Lagrange-Newton Method; 5.2.2 Sequential Quadratic Programming (SQP); 5.3 Calculation of Derivatives for Reduced Discretization; 5.3.1 Sensitivity Equation Approach. 5.3.2 Adjoint Equation Approach: The Discrete Case. The intention of this textbook is to provide both, the theoretical and computational tools that are necessary to investigate and to solve optimal control problems with ordinary differential equations and differential-algebraic equations. An emphasis is placed on the interplay between the continuous optimal control problem, which typically is defined and analyzed in a Banach space setting, and discrete optimal control problems, which are obtained by discretization and lead to finite dimensional optimization problems. Print version record. Includes bibliographical references and index. English. Control theory Mathematical models. Mathematical optimization. http://id.loc.gov/authorities/subjects/sh85082127 Théorie de la commande Modèles mathématiques. Optimisation mathématique. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Control theory Mathematical models fast Mathematical optimization fast has work: Optimal control of ODEs and DAEs (Text) https://id.oclc.org/worldcat/entity/E39PCFvppCbFCwwkgRrWHhTT73 https://id.oclc.org/worldcat/ontology/hasWork Print version: Gerdts, Matthias. Optimal Control of ODEs and DAEs. Berlin : De Gruyter, ©2012 9783110249958 De Gruyter textbook. FWS01 ZDB-4-EBU FWS_PDA_EBU https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=430058 Volltext FWS01 ZDB-4-EBU FWS_PDA_EBU https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=3689087 Volltext |
| spellingShingle | Gerdts, Matthias Optimal Control of ODEs and DAEs. De Gruyter textbook. Preface; 1 Introduction; 1.1 DAE Optimal Control Problems; 1.1.1 Perturbation Index; 1.1.2 Consistent Initial Values; 1.1.3 Index Reduction and Stabilization; 1.2 Transformation Techniques; 1.2.1 Transformation to Fixed Time Interval; 1.2.2 Transformation to Autonomous Problem; 1.2.3 Transformation of Tschebyscheff Problems; 1.2.4 Transformation of L 1 -Minimization Problems; 1.2.5 Transformation of Interior-Point Constraints; 1.3 Overview; 1.4 Exercises; 2 Infinite Optimization Problems; 2.1 Function Spaces; 2.1.1 Topological Spaces, Banach Spaces, and Hilbert Spaces. 2.1.2 Mappings and Dual Spaces2.1.3 Derivatives, Mean-Value Theorem, and Implicit Function Theorem; 2.1.4 Lp-Spaces, Wq; P-Spaces, Absolutely Continuous Functions, Functions of Bounded Variation; 2.2 The DAE Optimal Control Problem as an Infinite Optimization Problem; 2.3 Necessary Conditions for Infinite Optimization Problems; 2.3.1 Existence of a Solution; 2.3.2 Conic Approximation of Sets; 2.3.3 Separation Theorems; 2.3.4 First Order Necessary Optimality Conditions of Fritz John Type; 2.3.5 Constraint Qualifications and Karush-Kuhn-Tucker Conditions; 2.4 Exercises. 3 Local Minimum Principles3.1 Problems without Pure State and Mixed Control-State Constraints; 3.1.1 Representation of Multipliers; 3.1.2 Local Minimum Principle; 3.1.3 Constraint Qualifications and Regularity; 3.2 Problems with Pure State Constraints; 3.2.1 Representation of Multipliers; 3.2.2 Local Minimum Principle; 3.2.3 Finding Controls on Active State Constraint Arcs; 3.2.4 Jump Conditions for the Adjoint; 3.3 Problems with Mixed Control-State Constraints; 3.3.1 Representation of Multipliers; 3.3.2 Local Minimum Principle; 3.4 Summary of Local Minimum Principles for Index-One Problems. 3.5 Exercises4 Discretization Methods for ODEs and DAEs; 4.1 Discretization by One-Step Methods; 4.1.1 The Euler Method; 4.1.2 Runge-Kutta Methods; 4.1.3 General One-Step Method; 4.1.4 Consistency, Stability, and Convergence of One-Step Methods; 4.2 Backward Differentiation Formulas (BDF); 4.3 Linearized Implicit Runge-Kutta Methods; 4.4 Automatic Step-size Selection; 4.5 Computation of Consistent Initial Values; 4.5.1 Projection Method for Consistent Initial Values; 4.5.2 Consistent Initial Values via Relaxation; 4.6 Shooting Techniques for Boundary Value Problems. 4.6.1 Single Shooting Method using Projections4.6.2 Single Shooting Method using Relaxations; 4.6.3 Multiple Shooting Method; 4.7 Exercises; 5 Discretization of Optimal Control Problems; 5.1 Direct Discretization Methods; 5.1.1 Full Discretization Approach; 5.1.2 Reduced Discretization Approach; 5.1.3 Control Discretization; 5.2 A Brief Introduction to Sequential Quadratic Programming; 5.2.1 Lagrange-Newton Method; 5.2.2 Sequential Quadratic Programming (SQP); 5.3 Calculation of Derivatives for Reduced Discretization; 5.3.1 Sensitivity Equation Approach. Control theory Mathematical models. Mathematical optimization. http://id.loc.gov/authorities/subjects/sh85082127 Théorie de la commande Modèles mathématiques. Optimisation mathématique. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Control theory Mathematical models fast Mathematical optimization fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85082127 |
| title | Optimal Control of ODEs and DAEs. |
| title_auth | Optimal Control of ODEs and DAEs. |
| title_exact_search | Optimal Control of ODEs and DAEs. |
| title_full | Optimal Control of ODEs and DAEs. |
| title_fullStr | Optimal Control of ODEs and DAEs. |
| title_full_unstemmed | Optimal Control of ODEs and DAEs. |
| title_short | Optimal Control of ODEs and DAEs. |
| title_sort | optimal control of odes and daes |
| topic | Control theory Mathematical models. Mathematical optimization. http://id.loc.gov/authorities/subjects/sh85082127 Théorie de la commande Modèles mathématiques. Optimisation mathématique. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Control theory Mathematical models fast Mathematical optimization fast |
| topic_facet | Control theory Mathematical models. Mathematical optimization. Théorie de la commande Modèles mathématiques. Optimisation mathématique. MATHEMATICS Calculus. MATHEMATICS Mathematical Analysis. Control theory Mathematical models Mathematical optimization |
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