Ebook: Optimization—Theory and Applications: Problems with Ordinary Differential Equations

This book has grown out of lectures and courses in calculus of variations and optimization taught for many years at the University of Michigan to graduate students at various stages of their careers, and always to a mixed audience of students in mathematics and engineering. It attempts to present a balanced view of the subject, giving some emphasis to its connections with the classical theory and to a number of those problems of economics and engineering which have motivated so many of the present developments, as well as presenting aspects of the current theory, particularly value theory and existence theorems. However, the presentation ofthe theory is connected to and accompanied by many concrete problems of optimization, classical and modern, some more technical and some less so, some discussed in detail and some only sketched or proposed as exercises. No single part of the subject (such as the existence theorems, or the more traditional approach based on necessary conditions and on sufficient conditions, or the more recent one based on value function theory) can give a sufficient representation of the whole subject. This holds particularly for the existence theorems, some of which have been conceived to apply to certain large classes of problems of optimization. For all these reasons it is essential to present many examples (Chapters 3 and 6) before the existence theorems (Chapters 9 and 11-16), and to investigate these examples by means of the usual necessary conditions, sufficient conditions, and value function theory.

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Content:
Front Matter....Pages i-xv
Problems of Optimization—A General View....Pages 1-23
The Classical Problems of the Calculus of Variations: Necessary Conditions and Sufficient Conditions; Convexity and Lower Semicontinuity....Pages 24-115
Examples and Exercises on Classical Problems....Pages 116-158
Statement of the Necessary Condition for Mayer Problems of Optimal Control....Pages 159-195
Lagrange and Bolza Problems of Optimal Control and Other Problems....Pages 196-205
Examples and Exercises on Optimal Control....Pages 206-232
Proofs of the Necessary Condition for Control Problems and Related Topics....Pages 233-270
The Implicit Function Theorem and the Elementary Closure Theorem....Pages 271-308
Existence Theorems: The Bounded, or Elementary, Case....Pages 309-324
Closure and Lower Closure Theorems under Weak Convergence....Pages 325-366
Existence Theorems: Weak Convergence and Growth Conditions....Pages 367-402
Existence Theorems: The Case of an Exceptional Set of No Growth....Pages 403-416
Existence Theorems: The Use of Lipschitz and Tempered Growth Conditions....Pages 417-429
Existence Theorems: Problems of Slow Growth....Pages 430-442
Existence Theorems: The Use of Mere Pointwise Convergence on the Trajectories....Pages 443-452
Existence Theorems: Problems with No Convexity Assumptions....Pages 453-473
Duality and Upper Semicontinuity of Set Valued Functions....Pages 474-502
Approximation of Usual and of Generalized Solutions....Pages 503-521
Back Matter....Pages 523-542

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Content:
Front Matter....Pages i-xv
Problems of Optimization—A General View....Pages 1-23
The Classical Problems of the Calculus of Variations: Necessary Conditions and Sufficient Conditions; Convexity and Lower Semicontinuity....Pages 24-115
Examples and Exercises on Classical Problems....Pages 116-158
Statement of the Necessary Condition for Mayer Problems of Optimal Control....Pages 159-195
Lagrange and Bolza Problems of Optimal Control and Other Problems....Pages 196-205
Examples and Exercises on Optimal Control....Pages 206-232
Proofs of the Necessary Condition for Control Problems and Related Topics....Pages 233-270
The Implicit Function Theorem and the Elementary Closure Theorem....Pages 271-308
Existence Theorems: The Bounded, or Elementary, Case....Pages 309-324
Closure and Lower Closure Theorems under Weak Convergence....Pages 325-366
Existence Theorems: Weak Convergence and Growth Conditions....Pages 367-402
Existence Theorems: The Case of an Exceptional Set of No Growth....Pages 403-416
Existence Theorems: The Use of Lipschitz and Tempered Growth Conditions....Pages 417-429
Existence Theorems: Problems of Slow Growth....Pages 430-442
Existence Theorems: The Use of Mere Pointwise Convergence on the Trajectories....Pages 443-452
Existence Theorems: Problems with No Convexity Assumptions....Pages 453-473
Duality and Upper Semicontinuity of Set Valued Functions....Pages 474-502
Approximation of Usual and of Generalized Solutions....Pages 503-521
Back Matter....Pages 523-542
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