Berkovitz / Medhin Nonlinear Optimal Control Theory

1. Auflage 2013
ISBN: 978-1-4665-6027-7
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 392 Seiten

Reihe: Chapman & Hall/CRC Applied Mathematics & Nonlinear Science

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.

Zielgruppe

Mathematicians, engineers, graduate students in mathematics, and researchers who use the techniques of optimal control.

Autoren/Hrsg.

Berkovitz, Leonard David

Medhin, Negash G.

Fachgebiete

• Technische Wissenschaften Elektronik | Nachrichtentechnik Nachrichten- und Kommunikationstechnik Regelungstechnik
• Mathematik | Informatik Mathematik Operations Research

Inhaltsverzeichnis

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 e-Problem

The e-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

e-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