Calculus of Variations and Optimal Control Theory A Concise Introduction

This textbook offers a concise yet rigorous introduction to calculus of variations and optimal control theory, and is a self-contained resource for graduate students in engineering, applied mathematics, and related subjects. Designed specifically for a one-semester course, the book begins with calcu...

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Bibliographic Details
Main Author Liberzon, Daniel
Format eBook Book
LanguageEnglish
Published Princeton Princeton University Press 2011
Edition1
Subjects
Calculus of variations
Calculus of variations.-fast
Control theory
Control Theory & Optimization
Control theory.-fast
General Engineering & Project Administration
General References
General Topics for Engineers
MATHEMATICS
MATHEMATICS / Applied
MATHEMATICS / Calculus
MATHEMATICS / Mathematical Analysis
MATHEMATICS / Optimization
Optimale Kontrolle.-gnd
Process Design, Control & Automation
Variationsrechnung.-gnd
Online AccessGet full text
ISBN0691151873
9780691151878
1400842646
9781400842643
DOI10.1515/9781400842643

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Table of Contents:
  • Title Page Preface Table of Contents 1. Introduction 2. Calculus of Variations 3. From Calculus of Variations to Optimal Control 4. The Maximum Principle 5. The Hamilton-Jacobi-Bellman Equation 6. The Linear Quadratic Regulator 7. Advanced Topics Bibliography Index
  • Cover -- Title Page -- Copyright Page -- Table of Contents -- Preface -- Chapter 1. Introduction -- 1.1 Optimal control problem -- 1.2 Some background on finite-dimensional optimization -- 1.2.1 Unconstrained optimization -- 1.2.2 Constrained optimization -- 1.3 Preview of infinite-dimensional optimization -- 1.3.1 Function spaces, norms, and local minima -- 1.3.2 First variation and first-order necessary condition -- 1.3.3 Second variation and second-order conditions -- 1.3.4 Global minima and convex problems -- 1.4 Notes and references for Chapter 1 -- Chapter 2. Calculus of Variations -- 2.1 Examples of variational problems -- 2.1.1 Dido's isoperimetric problem -- 2.1.2 Light reflection and refraction -- 2.1.3 Catenary -- 2.1.4 Brachistochrone -- 2.2 Basic calculus of variations problem -- 2.2.1 Weak and strong extrema -- 2.3 First-order necessary conditions for weak extrema -- 2.3.1 Euler-Lagrange equation -- 2.3.2 Historical remarks -- 2.3.3 Technical remarks -- 2.3.4 Two special cases -- 2.3.5 Variable-endpoint problems -- 2.4 Hamiltonian formalism and mechanics -- 2.4.1 Hamilton's canonical equations -- 2.4.2 Legendre transformation -- 2.4.3 Principle of least action and conservation laws -- 2.5 Variational problems with constraints -- 2.5.1 Integral constraints -- 2.5.2 Non-integral constraints -- 2.6 Second-order conditions -- 2.6.1 Legendre's necessary condition for a weak minimum -- 2.6.2 Sufficient condition for a weak minimum -- 2.7 Notes and references for Chapter 2 -- Chapter 3. From Calculus of Variations to Optimal Control -- 3.1 Necessary conditions for strong extrema -- 3.1.1 Weierstrass-Erdmann corner conditions -- 3.1.2 Weierstrass excess function -- 3.2 Calculus of variations versus optimal control -- 3.3 Optimal control problem formulation and assumptions -- 3.3.1 Control system -- 3.3.2 Cost functional -- 3.3.3 Target set
  • 3.4 Variational approach to the fixed-time, free-endpoint problem -- 3.4.1 Preliminaries -- 3.4.2 First variation -- 3.4.3 Second variation -- 3.4.4 Some comments -- 3.4.5 Critique of the variational approach and preview of the maximum principle -- 3.5 Notes and references for Chapter 3 -- Chapter 4. The Maximum Principle -- 4.1 Statement of the maximum principle -- 4.1.1 Basic fixed-endpoint control problem -- 4.1.2 Basic variable-endpoint control problem -- 4.2 Proof of the maximum principle -- 4.2.1 From Lagrange to Mayer form -- 4.2.2 Temporal control perturbation -- 4.2.3 Spatial control perturbation -- 4.2.4 Variational equation -- 4.2.5 Terminal cone -- 4.2.6 Key topological lemma -- 4.2.7 Separating hyperplane -- 4.2.8 Adjoint equation -- 4.2.9 Properties of the Hamiltonian -- 4.2.10 Transversality condition -- 4.3 Discussion of the maximum principle -- 4.3.1 Changes of variables -- 4.4 Time-optimal control problems -- 4.4.1 Example: double integrator -- 4.4.2 Bang-bang principle for linear systems -- 4.4.3 Nonlinear systems, singular controls, and Lie brackets -- 4.4.4 Fuller's problem -- 4.5 Existence of optimal controls -- 4.6 Notes and references for Chapter 4 -- Chapter 5. The Hamilton-Jacobi-Bellman Equation -- 5.1 Dynamic programming and the HJB equation -- 5.1.1 Motivation: the discrete problem -- 5.1.2 Principle of optimality -- 5.1.3 HJB equation -- 5.1.4 Sufficient condition for optimality -- 5.1.5 Historical remarks -- 5.2 HJB equation versus the maximum principle -- 5.2.1 Example: nondifferentiable value function -- 5.3 Viscosity solutions of the HJB equation -- 5.3.1 One-sided differentials -- 5.3.2 Viscosity solutions of PDEs -- 5.3.3 HJB equation and the value function -- 5.4 Notes and references for Chapter 5 -- Chapter 6. The Linear Quadratic Regulator -- 6.1 Finite-horizon LQR problem -- 6.1.1 Candidate optimal feedback law
  • 6.1.2 Riccati differential equation -- 6.1.3 Value function and optimality -- 6.1.4 Global existence of solution for the RDE -- 6.2 Infinite-horizon LQR problem -- 6.2.1 Existence and properties of the limit -- 6.2.2 Infinite-horizon problem and its solution -- 6.2.3 Closed-loop stability -- 6.2.4 Complete result and discussion -- 6.3 Notes and references for Chapter 6 -- Chapter 7. Advanced Topics -- 7.1 Maximum principle on manifolds -- 7.1.1 Differentiable manifolds -- 7.1.2 Re-interpreting the maximum principle -- 7.1.3 Symplectic geometry and Hamiltonian ows -- 7.2 HJB equation, canonical equations, and characteristics -- 7.2.1 Method of characteristics -- 7.2.2 Canonical equations as characteristics of the HJB equation -- 7.3 Riccati equations and inequalities in robust control -- 7.3.1 L2 gain -- 7.3.2 H** control problem -- 7.3.3 Riccati inequalities and LMIs -- 7.4 Maximum principle for hybrid control systems -- 7.4.1 Hybrid optimal control problem -- 7.4.2 Hybrid maximum principle -- 7.4.3 Example: light reflection -- 7.5 Notes and references for Chapter 7 -- Bibliography -- Index
  • Chapter Seven. Advanced Topics --
  • Contents --
  • Chapter One. Introduction --
  • Chapter Five. The Hamilton-Jacobi-Bellman Equation --
  • Chapter Two. Calculus of Variations --
  • Chapter Four. The Maximum Principle --
  • Preface --
  • Frontmatter --
  • Index
  • Chapter Six. The Linear Quadratic Regulator --
  • Bibliography --
  • Chapter Three. From Calculus of Variations to Optimal Control --