Principles of Dynamic Optimization

✍ Scribed by Piernicola Bettiol , Richard B. Vinter

• Publisher: Springer Nature Switzerland
• Year: 2024
• Tongue: English
• Leaves: 789
• Series: Springer Monographs in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

This monograph explores key principles in the modern theory of dynamic optimization, incorporating important advances in the field to provide a comprehensive, mathematically rigorous reference. Emphasis is placed on nonsmooth analytic techniques, and an in-depth treatment of necessary conditions, minimizer regularity, and global optimality conditions related to the Hamilton-Jacobi equation is given. New, streamlined proofs of fundamental theorems are incorporated throughout the text that eliminate earlier, cumbersome reductions and constructions. The first chapter offers an extended overview of dynamic optimization and its history that details the shortcomings of the elementary theory and demonstrates how a deeper analysis aims to overcome them. Aspects of dynamic programming well-matched to analytical techniques are considered in the final chapter, including characterization of extended-value functions associated with problems having endpoint and state constraints, inverse verification theorems, sensitivity relationships, and links to the maximum principle.

This text will be a valuable resource for those seeking an understanding of dynamic optimization. The lucid exposition, insights into the field, and comprehensive coverage will benefit postgraduates, researchers, and professionals in system science, control engineering, optimization, and applied mathematics.

• Discusses numerous recent advances not previously available in book form
• Includes a comprehensive treatment of dynamic programming based on a study of system invariance
• Provides a self-contained exposition of non-smooth analysis, emphasizing aspects relevant to optimization

✦ Table of Contents

Preface
Contents
Notation
1 Overview
1.1 Dynamic Optimization
1.2 The Calculus of Variations
1.3 Existence of Minimizers and Tonelli's Direct Method
1.4 Sufficient Conditions and the Hamilton Jacobi Equation
1.5 The Maximum Principle
1.6 Dynamic Programming
1.7 Nonsmoothness
1.8 Nonsmooth Analysis
1.9 Nonsmooth Dynamic Optimization
1.10 Epilogue
1.11 Appendix: Proof of the Classical Maximum Principle
1.12 Exercises
1.13 Notes for Chapter 1
2 Set Convergence, Measurability and Existence of Minimizers
2.1 Introduction
2.2 Convergence of Sets and Continuity of Multifunctions
2.3 Measurable Multifunctions
2.4 The Generalized Bolza Problem
2.5 Exercises
2.6 Notes for Chapter 2
3 Variational Principles
3.1 Introduction
3.2 Exact Penalization
3.3 Ekeland's Theorem
3.4 Quadratic Inf Convolution
3.5 Variational Principles with Smooth Perturbation Terms
3.6 Mini-Max Theorems
3.7 Exercises
3.8 Notes for Chapter 3
4 Nonsmooth Analysis
4.1 Introduction
4.2 Normal Cones
4.3 Subdifferentials
4.4 Difference Quotient Representations
4.5 Nonsmooth Mean Value Inequalities
4.6 Characterization of Limiting Subgradients
4.7 Subgradients of Lipschitz Continuous Functions
4.8 The Distance Function
4.9 Criteria for Lipschitz Continuity
4.10 Relations Between Normal and Tangent Cones
4.11 Interior of Clarke's Tangent Cone
4.12 Appendix: Proximal Analysis in Hilbert Space
4.13 Exercises
4.14 Notes for Chapters 4 and 5
5 Subdifferential Calculus
5.1 Introduction
5.2 A Marginal Function Principle
5.3 Partial Limiting Subgradients
5.4 A Sum Rule
5.5 A Nonsmooth Chain Rule
5.6 Lagrange Multiplier Rules
5.7 Max Rule for an Infinite Family of Functions
5.8 Exercises
5.9 Notes for Chapter 5
6 Differential Inclusions
6.1 Introduction
6.2 Existence and Estimation of F Trajectories
6.3 Perturbed Differential Inclusions
6.4 Existence of Minimizing F Trajectories
6.5 Relaxation
6.6 Estimates on Trajectories Confined to a Closed Subset
6.7 Exercises
6.8 Notes for Chapter 6
7 The Maximum Principle
7.1 Introduction
7.2 Clarke's Nonsmooth Maximum Principle
7.3 A Preliminary Maximum Principle, for Dynamic Optimization Problems with no End-Point Constraints
7.4 Proof of Theorem7.2.1
7.5 Exercises
7.6 Notes for Chapter 7
8 The Generalized Euler-Lagrange and Hamiltonian Inclusion Conditions
8.1 Introduction
8.2 Pseudo Lipschitz Continuity
8.3 Unbounded Differential Inclusions
8.4 The Generalized Euler Lagrange Condition
8.5 Special Cases
8.6 Proof of Theorem 8.4.3
8.7 The Hamiltonian Inclusion for Convex Velocity Sets
8.8 The Hamiltonian Inclusion for Non-convex Velocity Sets
8.9 Discussion and a Counter-Example
8.10 Appendix: Dualization of the Euler Lagrange Inclusion
8.11 Exercises
8.12 Notes for Chapter 8
9 Free End-Time Problems
9.1 Introduction
9.2 Lipschitz Time Dependence
9.3 Essential Values
9.4 Measurable Time Dependence
9.5 Proof of Theorem 9.4.1
9.6 A Free End-Time Maximum Principle
9.7 Appendix: Metrics on the Space of FreeEnd-Time Trajectories
9.8 Exercises
9.9 Notes for Chapter 9
10 The Maximum Principle for Problems with Pathwise Constraints
10.1 Introduction
10.2 Problems with Pure State Constraints: Preliminary Discussion
10.3 Convergence of Measures
10.4 The Maximum Principle (Pure State Constraints)
10.5 Proof of Theorem 10.4.1
10.6 Maximum Principles for Free End-Time Problems with State Constraints
10.7 Non-degenerate Conditions
10.8 Mixed Constraints
10.9 Exercises
10.10 Notes for Chapter 10
11 The Euler-Lagrange and Hamiltonian Inclusion Conditions in the Presence of State Constraints
11.1 Introduction
11.2 The Euler Lagrange Inclusion
11.3 Proof of Theorem11.2.1
11.4 Free End-Time Problems with State Constraints
11.5 Non-degenerate Necessary Conditions
11.6 Exercises
11.7 Notes for Chapter 11
12 Regularity of Minimizers
12.1 Introduction
12.2 Tonelli Regularity
12.3 Proof of The Generalized Tonelli Regularity Theorem
12.4 Lipschitz Continuous Minimizers
12.5 Nonautonomous Variational Problems with State Constraints
12.6 Bounded Controls
12.7 Lipschitz Continuous Controls
12.8 Exercises
12.9 Notes for Chapter 12
13 Dynamic Programming
13.1 Introduction
13.2 Invariance Theorems
13.3 The Value Function and Generalized Solutions of the Hamilton Jacobi Equation
13.4 Local Verification Theorems
13.5 Costate Trajectories and Gradients of the Value Function
13.6 State Constrained Problems
13.7 Proofs of Theorems 13.6.1 and 13.6.2
13.8 Costate Trajectories and Gradients of the Value Functions for State-Constrained Problems
13.9 Semiconcavity and the Value Function
13.10 The Infinite Horizon Problem
13.11 The Minimum Time Problem
13.12 Viscosity Solutions of the Hamilton Jacobi Equation
13.13 A Comparison Theorem for Viscosity Solutions
13.14 Exercises
13.15 Notes for Chapter 13
References
Index

✦ Subjects

Dynamic Optimization, Optimal Control, Nonsmooth Analysis, Differential Inclusions, Hamilton-Jacobi Theory