| Description |
1 online resource (468 pages). |
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text file |
| Series |
De Gruyter textbook
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De Gruyter textbook.
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| 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. |
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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. |
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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. |
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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. |
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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. |
| Note |
5.3.2 Adjoint Equation Approach: The Discrete Case. |
| Summary |
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. |
| Bibliography |
Includes bibliographical references and index. |
| Local Note |
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America |
| Subject |
Control theory -- Mathematical models.
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Control theory -- Mathematical models. |
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Mathematical optimization.
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Mathematical optimization. |
| Genre/Form |
Electronic books.
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| Other Form: |
Print version: Gerdts, Matthias. Optimal Control of ODEs and DAEs. Berlin : De Gruyter, ©2012 9783110249958 |
| ISBN |
9783110249996 (electronic book) |
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3110249995 (electronic book) |
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9783110249958 (alkaline paper) |
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