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Optimal Control
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783764340759.html
Optimal control provides a unified perspective of optimization problems, arising in scheduling and the control of engineering devices, that are beyond the reach of traditional analytical and computational techniques. In addition, the field of optimal control has contributed significant advances to branches of applied mathematics and broad applications in process control, scheduling, robotics, resource economics, and other areas. This text aims to bring together new results and advances in optimal control, providing a self-contained analysis and incorporating numerous simplifications and unifying features for the subjects' key concepts and foundations.
Preface xi
Notation xvii
Overview
1(60)
Optimal Control
1(3)
The Calculus of Variations
4(14)
Existence of Minimizers and Tonelli's Direct Method
18(3)
Sufficient Conditions and the Hamilton-Jacobi Equation
21(4)
The Maximum Principle
25(5)
Dynamic Programming
30(5)
Nonsmoothness
35(4)
Nonsmooth Analysis
39(13)
Nonsmooth Optimal Control
52(4)
Epilogue
56(4)
Notes for
Chapter 1
60(1)
Measurable Multifunctions and Differential Inclusions
61(48)
Introduction
61(1)
Convergence of Sets
62(2)
Measurable Multifunctions
64(11)
Existence and Estimation of F-Trajectories
75(11)
Perturbed Differential Inclusions
86(5)
Existence of Minimizing F-Trajectories
91(3)
Relaxation
94(6)
The Generalized Bolza Problem
100(8)
Notes for
Chapter 2
108(1)
Variational Principles
109(18)
Introduction
109(1)
Exact Penalization
110(1)
Ekeland's Theorem
111(4)
Mini-Max Theorems
115(10)
Notes for
Chapter 3
125(2)
Nonsmooth Analysis
127(52)
Introduction
127(1)
Normal Cones
128(5)
Subdifferentials
133(6)
Difference Quotient Representations
139(5)
Nonsmooth Mean Value Inequalities
144(5)
Characterization of Limiting Subgradients
149(5)
Subgradients of Lipschitz Continuous Functions
154(7)
The Distance Function
161(5)
Criteria for Lipschitz Continuity
166(4)
Relationships Between Normal and Tangent Cones
170(9)
Subdifferential Calculus
179(22)
Introduction
179(2)
A Marginal Function Principle
181(4)
Partial Limiting Subgradients
185(2)
A Sum Rule
187(3)
A Nonsmooth Chain Rule
190(3)
Lagrange Multiplier Rules
193(4)
Notes for
Chapters 4 and 5
197(4)
The Maximum Principle
201(32)
Introduction
201(2)
The Maximum Principle
203(5)
Derivation of the Maximum Principle from the Extended Euler Condition
208(6)
A Smooth Maximum Principle
214(14)
Notes for
Chapter 6
228(5)
The Extended Euler-Lagrange and Hamilton Conditions
233(52)
Introduction
233(4)
Properties of the Distance Function
237(5)
Necessary Conditions for a Finite Lagrangian Problem
242(10)
The Extended Euler-Lagrange Condition: Nonconvex Velocity Sets
252(7)
The Extended Euler-Lagrange Condition: Convex Velocity Sets
259(5)
Dualization of the Extended Euler-Lagrange Condition
264(13)
The Extended Hamilton Condition
277(3)
Notes for
Chapter 7
280(5)
Necessary Conditions for Free End-Time Problems
285(36)
Introduction
285(3)
Lipschitz Time Dependence
288(7)
Essential Values
295(2)
Measurable Time Dependence
297(4)
Proof of Theorem 8.4.1
301(9)
Proof of Theorem 8.4.2
310(3)
A Free End-Time Maximum Principle
313(5)
Notes for
Chapter 8
318(3)
The Maximum Principle for State Constrained Problems
321(40)
Introduction
321(3)
Convergence of Measures
324(5)
The Maximum Principle for Problems with State Constraints
329(5)
Derivation of the Maximum Principle for State Constrained Problems from the Euler-Lagrange Condition
334(5)
A Smooth Maximum Principle for State Constrained Problems
339(20)
Notes for
Chapter 9
359(2)
Differential Inclusions with State Constraints
361(36)
Introduction
361(1)
A Finite Lagrangian Problem
362(6)
The Extended Euler-Lagrange Condition for State Constrained Problems: Nonconvex Velocity Sets
368(7)
Necessary Conditions for State Constrained Problems: Convex Velocity Sets
375(7)
Free Time Problems with State Constraints
382(5)
Nondegenerate Necessary Conditions
387(9)
Notes for
Chapter 10
396(1)
Regularity of Minimizers
397(38)
Introduction
397(6)
Tonelli Regularity
403(5)
Proof of The Generalized Tonelli Regularity Theorem
408(9)
Lipschitz Continuous Minimizers
417(5)
Autonomous Variational Problems with State Constraints
422(3)
Bounded Controls
425(3)
Lipschitz Continuous Controls
428(4)
Notes for
Chapter 11
432(3)
Dynamic Programming
435(58)
Introduction
435(7)
Invariance Theorems
442(10)
The Value Function and Generalized Solutions of the Hamilton-Jacobi Equation
452(13)
Local Verification Theorems
465(9)
Adjoint Arcs and Gradients of the Value Function
474(9)
State Constrained Problems
483(4)
Notes for
Chapter 12
487(6)
References 493(12)
Index 505