Optimization-Theory and Applications

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