Optimization¿Theory and Applications

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.

1 Problems of Optimization―A General View.- 1.1 Classical Lagrange Problems of the Calculus of Variations.- 1.2 Classical Lagrange Problems with Constraints on the Derivatives.- 1.3 Classical Bolza Problems of the Calculus of Variations.- 1.4 Classical Problems Depending on Derivatives of Higher Order.- 1.5 Examples of Classical Problems of the Calculus of Variations.- 1.6 Remarks.- 1.7 The Mayer Problems of Optimal Control.- 1.8 Lagrange and Bolza Problems of Optimal Control.- 1.9 Theoretical Equivalence of Mayer, Lagrange, and Bolza Problems of Optimal Control. Problems of the Calculus of Variations as Problems of Optimal Control.- 1.10 Examples of Problems of Optimal Control.- 1.11 Exercises.- 1.12 The Mayer Problems in Terms of Orientor Fields.- 1.13 The Lagrange Problems of Control as Problems of the Calculus of Variations with Constraints on the Derivatives.- 1.14 Generalized Solutions.- Bibliographical Notes.- 2 The Classical Problems of the Calculus of Variations: Necessary Conditions and Sufficient Conditions; Convexity and Lower Semicontinuity.- 2.1 Minima and Maxima for Lagrange Problems of the Calculus of Variations.- 2.2 Statement of Necessary Conditions.- 2.3 Necessary Conditions in Terms of Gateau Derivatives.- 2.4 Proofs of the Necessary Conditions and of Their Invariant Character.- 2.5 Jacobi’s Necessary Condition.- 2.6 Smoothness Properties of Optimal Solutions.- 2.7 Proof of the Euler and DuBois-Reymond Conditions in the Unbounded Case.- 2.8 Proof of the Transversality Relations.- 2.9 The String Property and a Form of Jacobi’s Necessary Condition.- 2.10 An Elementary Proof of Weierstrass’s Necessary Condition.- 2.11 Classical Fields and Weierstrass’s Sufficient Conditions.- 2.12 More Sufficient Conditions.- 2.13 Value Function and Further Sufficient Conditions.- 2.14 Uniform Convergence and Other Modes of Convergence.- 2.15 Semicontinuity of Functionals.- 2.16 Remarks on Convex Sets and Convex Real Valued Functions.- 2.17 A Lemma Concerning Convex Integrands.- 2.18 Convexity and Lower Semicontinuity: A Necessary and Sufficient Condition.- 2.19 Convexity as a Necessary Condition for Lower Semicontinuity.- 2.20 Statement of an Existence Theorem for Lagrange Problems of the Calculus of Variations.- Bibliographical Notes.- 3 Examples and Exercises on Classical Problems.- 3.1 An Introductory Example.- 3.2 Geodesics.- 3.3 Exercises.- 3.4 Fermat’s Principle.- 3.5 The Ramsay Model of Economic Growth.- 3.6 Two Isoperimetric Problems.- 3.7 More Examples of Classical Problems.- 3.8 Miscellaneous Exercises.- 3.9 The Integral I = ?(x?2 ? x2)dt.- 3.10 The Integral I = ?xx?2dt.- 3.11 The Integral I = ?x?2(1 + x?)2dt.- 3.12 Brachistochrone, or Path of Quickest Descent.- 3.13 Surface of Revolution of Minimum Area.- 3.14 The Principles of Mechanics.- Bibliographical Notes.- 4 Statement of the Necessary Condition for Mayer Problems of Optimal Control.- 4.1 Some General Assumptions.- 4.2 The Necessary Condition for Mayer Problems of Optimal Control.- 4.3 Statement of an Existence Theorem for Mayer’s Problems of Optimal Control.- 4.4 Examples of Transversality Relations for Mayer Problems.- 4.5 The Value Function.- 4.6 Sufficient Conditions.- 4.7 Appendix: Derivation of Some of the Classical Necessary Conditions of Section 2.1 from the Necessary Condition for Mayer Problems of Optimal Control.- 4.8 Appendix: Derivation of the Classical Necessary Condition for Isoperimetric Problems from the Necessary Condition for Mayer Problems of Optimal Control.- 4.9 Appendix: Derivation of the Classical Necessary Condition for Lagrange Problems of the Calculus of Variations with Differential Equations as Constraints.- Bibliographical Notes.- 5 Lagrange and Bolza Problems of Optimal Control and Other Problems.- 5.1 The Necessary Condition for Bolza and Lagrange Problems of Optimal Control.- 5.2 Derivation of Properties (P1?)―(P4?) from (P1)―(P4).- 5.3 Examples of Applications of the Necessary Conditions for Lagrange Problems of Optimal Control.- 5.4 The Value Function.- 5.5 Sufficient Conditions for the Bolza Problem.- Bibliographical Notes.- 6 Examples and Exercises on Optimal Control.- 6.1 Stabilization of a Material Point Moving on a Straight Line under a Limited External Force.- 6.2 Stabilization of a Material Point under an Elastic Force and a Limited External Force.- 6.3 Minimum Time Stabilization of a Reentry Vehicle.- 6.4 Soft Landing on the Moon.- 6.5 Three More Problems on the Stabilization of a Point Moving on a Straight Line.- 6.6 Exercises.- 6.7 Optimal Economic Growth.- 6.8 Two More Classical Problems.- 6.9 The Navigation Problem.- Bibliographical Notes.- 7 Proofs of the Necessary Condition for Control Problems and Related Topics.- 7.1 Description of the Problem of Optimization.- 7.2 Sketch of the Proofs.- 7.3 The First Proof.- 7.4 Second Proof of the Necessary Condition.- 7.5 Proof of Boltyanskii’s Statements (4.6.iv-v).- Bibliographical Notes.- 8 The Implicit Function Theorem and the Elementary Closure Theorem.- 8.1 Remarks on Semicontinuous Functionals.- 8.2 The Implicit Function Theorem.- 8.3 Selection Theorems.- 8.4 Convexity, Carathéodory’s Theorem, Extreme Points.- 8.5 Upper Semicontinuity Properties of Set Valued Functions.- 8.6 The Elementary Closure Theorem.- 8.7 Some Fatou-Like Lemmas.- 8.8 Lower Closure Theorems with Respect to Uniform Convergence.- Bibliographical Notes.- 9 Existence Theorems: The Bounded, or Elementary, Case.- 9.1 Ascoli’s Theorem.- 9.2 Filippov’s Existence Theorem for Mayer Problems of Optimal Control.- 9.3 Filippov’s Existence Theorem for Lagrange and Bolza Problems of Optimal Control.- 9.4 Elimination of the Hypothesis that A Is Compact in Filippov’s Theorem for Mayer Problems.- 9.5 Elimination of the Hypothesis that A Is Compact in Filippov’s Theorem for Lagrange and Bolza Problems.- 9.6 Examples.- Bibliographical Notes.- 10 Closure and Lower Closure Theorems under Weak Convergence.- 10.1 The Banach-Saks-Mazur Theorem.- 10.2 Absolute Integrability and Related Concepts.- 10.3 An Equivalence Theorem.- 10.4 A Few Remarks on Growth Conditions.- 10.5 The Growth Property (?) Implies Property (Q).- 10.6 Closure Theorems for Orientor Fields Based on Weak Convergence.- 10.7 Lower Closure Theorems for Orientor Fields Based on Weak Convergence.- 10.8 Lower Semicontinuity in the Topology of Weak Convergence.- 10.9 Necessary and Sufficient Conditions for Lower Closure.- Bibliographical Notes.- 11 Existence Theorems: Weak Convergence and Growth Conditions.- 11.1 Existence Theorems for Orientor Fields and Extended Problems.- 112 Elimination of the Hypothesis that A Is Bounded in Theorems (11.1. i-iv).- 11.3 Examples.- 11.4 Existence Theorems for Problems of Optimal Control with Unbounded Strategies.- 11.5 Elimination of the Hypothesis that A Is Bounded in Theorems (11.4.i-v).- 11.6 Examples.- 11.7 Counterexamples.- Bibliographical Notes.- 12 Existence Theorems: The Case of an Exceptional Set of No Growth.- 12.1 The Case of No Growth at the Points of a Slender Set. Lower Closure Theorems..- 12.2 Existence Theorems for Extended Free Problems with an Exceptional Slender Set.- 12.3 Existence Theorems for Problems of Optimal Control with an Exceptional Slender Set.- 12.4 Examples.- 12.5 Counterexamples.- Bibliographical Notes.- 13 Existence Theorems: The Use of Lipschitz and Tempered Growth Conditions.- 13.1 An Existence Theorem under Condition (D).- 13.2 Conditions of the F, G, and H Types Each Implying Property (D) and Weak Property (Q).- 13.3 Examples.- Bibliographical Notes.- 14 Existence Theorems: Problems of Slow Growth.- 14.1 Parametric Curves and Integrals.- 14.2 Transformation of Nonparametric into Parametric Integrals.- 14.3 Existence Theorems for (Nonparametric) Problems of Slow Growth.- 14.4 Examples.- Bibliographical Notes.- 15 Existence Theorems: The Use of Mere Pointwise Convergence on the Trajectories.- 15.1 The Helly Theorem.- 15.2 Closure Theorems with Components Converging Only Pointwise.- 15.3 Existence Theorems for Extended Problems Based on Pointwise Convergence.- 15.4 Existence Theorems for Problems of Optimal Control Based on Pointwise Convergence.- 15.5 Exercises.- Bibliographical Notes.- 16 Existence Theorems: Problems with No Convexity Assumptions.- 16.1 Lyapunov Type Theorems.- 16.2 The Neustadt Theorem for Mayer Problems with Bounded Controls.- 16.3 The Bang-Bang Theorem.- 16.4 The Neustadt Theorem for Lagrange and Bolza Problems with Bounded Controls.- 16.5 The Case of Unbounded Controls.- 16.6 Examples for the Unbounded Case.- 16.7 Problems of the Calculus of Variations without Convexity Assumptions.- Bibliographical Notes.- 17 Duality and Upper Semicontinuity of Set Valued Functions.- 17.1 Convex Functions on a Set.- 17.2 The Function T(x; z).- 17.3 Seminormality.- 17.4 Criteria for Property (Q).- 17.5 A Characterization of Property (Q) for the Sets $$\tilde Q$$(t, x) in Terms of Seminormality.- 17.6 Duality and Another Characterization of Property (Q) in Terms of Duality.- 17.7 Characterization of Optimal Solutions in Terms of Duality.- 17.8 Property (Q) as an Extension of Maximal Monotonicity.- Bibliographical Notes.- 18 Approximation of Usual and of Generalized Solutions.- 18.1 The Gronwall Lemma.- 18.2 Approximation of AC Solutions by Means of C1 Solutions.- 18.3 The Brouwer Fixed Point Theorem.- 18.4 Further Results Concerning the Approximation of AC Trajectories by Means of C1 Trajectories.- 18.5 The Infimum for AC Solutions Can Be Lower than the One for C1 Solutions.- 18.6 Approximation of Generalized Solutions by Means of Usual Solutions.- 18.7 The Infimum for Generalized Solutions Can Be Lower than the One for Usual Solutions.- Bibliographical Notes.- Author Index.