Nonlinear Optimal Control Theory

Nonlinear Optimal Control Theory

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Features

  • Covers the main topics of optimal control theory, including the Pontryagin principle, Bellman’s dynamic programming method, and theorems about the existence of optimal controls
  • Presents diverse examples from flight mechanics, chemical and electrical engineering, production planning models, and the classical calculus of variations
  • Describes control system models with and without time delays
  • Presents proofs of the Pontryagin maximum principle
  • Includes an introduction to Hamilton-Jacobi theory, tying together the viscosity solution and Subbotin approaches

Summary

Nonlinear Optimal Control Theory presents a deep, wide-ranging introduction to the mathematical theory of the optimal control of processes governed by ordinary differential equations and certain types of differential equations with memory. Many examples illustrate the mathematical issues that need to be addressed when using optimal control techniques in diverse areas.

Drawing on classroom-tested material from Purdue University and North Carolina State University, the book gives a unified account of bounded state problems governed by ordinary, integrodifferential, and delay systems. It also discusses Hamilton-Jacobi theory. By providing a sufficient and rigorous treatment of finite dimensional control problems, the book equips readers with the foundation to deal with other types of control problems, such as those governed by stochastic differential equations, partial differential equations, and differential games.

Table of Contents

Examples of Control Problems
Introduction
A Problem of Production Planning
Chemical Engineering
Flight Mechanics
Electrical Engineering
The Brachistochrone Problem
An Optimal Harvesting Problem
Vibration of a Nonlinear Beam

Formulation of Control Problems
Introduction
Formulation of Problems Governed by Ordinary Differential Equations
Mathematical Formulation
Equivalent Formulations
Isoperimetric Problems and Parameter Optimization
Relationship with the Calculus of Variations
Hereditary Problems

Relaxed Controls
Introduction
The Relaxed Problem; Compact Constraints
Weak Compactness of Relaxed Controls
Filippov’s Lemma
The Relaxed Problem; Non-Compact Constraints
The Chattering Lemma; Approximation to Relaxed Controls

Existence Theorems; Compact Constraints
Introduction
Non-Existence and Non-Uniqueness of Optimal Controls
Existence of Relaxed Optimal Controls
Existence of Ordinary Optimal Controls
Classes of Ordinary Problems Having Solutions
Inertial Controllers
Systems Linear in the State Variable

Existence Theorems; Non Compact Constraints
Introduction
Properties of Set Valued Maps
Facts from Analysis
Existence via the Cesari Property
Existence without the Cesari Property
Compact Constraints Revisited

The Maximum Principle and Some of its Applications
Introduction
A Dynamic Programming Derivation of the Maximum Principle
Statement of Maximum Principle
An Example
Relationship with the Calculus of Variations
Systems Linear in the State Variable
Linear Systems
The Linear Time Optimal Problem
Linear Plant-Quadratic Criterion Problem

Proof of the Maximum Principle
Introduction
Penalty Proof of Necessary Conditions in Finite Dimensions
The Norm of a Relaxed Control; Compact Constraints
Necessary Conditions for an Unconstrained Problem
The ε-Problem
The ε-Maximum Principle
The Maximum Principle; Compact Constraints
Proof of Theorem 6.3.9
Proof of Theorem 6.3.12
Proof of Theorem 6.3.17 and Corollary 6.3.19
Proof of Theorem 6.3.22

Examples
Introduction
The Rocket Car
A Non-Linear Quadratic Example
A Linear Problem with Non-Convex Constraints
A Relaxed Problem
The Brachistochrone Problem
Flight Mechanics
An Optimal Harvesting Problem
Rotating Antenna Example

Systems Governed by Integrodifferential Systems
Introduction
Problem Statement
Systems Linear in the State Variable
Linear Systems/The Bang-Bang Principle
Systems Governed by Integrodifferential Systems
Linear Plant Quadratic Cost Criterion
A Minimum Principle

Hereditary Systems
Introduction
Problem Statement and Assumptions
Minimum Principle
Some Linear Systems
Linear Plant-Quadratic Cost
Infinite Dimensional Setting

Bounded State Problems
Introduction
Statement of the Problem
ε-Optimality Conditions
Limiting Operations
The Bounded State Problem for Integrodifferential Systems
The Bounded State Problem for Ordinary Differential Systems
Further Discussion of the Bounded State Problem
Sufficiency Conditions
Nonlinear Beam Problem

Hamilton-Jacobi Theory
Introduction
Problem Formulation and Assumptions
Continuity of the Value Function
The Lower Dini Derivate Necessary Condition
The Value as Viscosity Solution
Uniqueness
The Value Function as Verification Function
Optimal Synthesis
The Maximum Principle

Bibliography

Index

Editorial Reviews

This book provides a thorough introduction to optimal control theory for nonlinear systems. … The book is enhanced by the inclusion of many examples, which are analyzed in detail using Pontryagin’s principle. … An important feature of the book is its systematic use of a relaxed control formulation of optimal control problems. …
—From the Foreword by Wendell Fleming

… more than a very useful research account and a handy reference to users of the theory-they also make it a pleasant and helpful study opportunity to students and other newcomers to the theory of optimal control.
—Zvi Artstein, in Mathematical Reviews

 
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