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- Introduces a variety of contemporary applications from control theory to numerical analysis
- Assumes a basic background in differential equations and advanced calculus, making the text suitable for beginning graduate students in mathematics and engineering
- Presents a complete treatment of the simplest problem in the calculus of variations
- Describes extensions and generalizations to vector and higher dimensional problems
- Covers optimal control, the maximum principle, linear optimal control, and feedback design

**Introduction to the Calculus of Variations and Control with Modern Applications** provides the fundamental background required to develop rigorous necessary conditions that are the starting points for theoretical and numerical approaches to modern variational calculus and control problems. The book also presents some classical sufficient conditions and discusses the importance of distinguishing between the necessary and sufficient conditions.

In the first part of the text, the author develops the calculus of variations and provides complete proofs of the main results. He explains how the ideas behind the proofs are essential to the development of modern optimization and control theory. Focusing on optimal control problems, the second part shows how optimal control is a natural extension of the classical calculus of variations to more complex problems.

By emphasizing the basic ideas and their mathematical development, this book gives you the foundation to use these mathematical tools to then tackle new problems. The text moves from simple to more complex problems, allowing you to see how the fundamental theory can be modified to address more difficult and advanced challenges. This approach helps you understand how to deal with future problems and applications in a realistic work environment.

*Calculus of Variations ***Historical Notes on the Calculus of Variations **

Some Typical Problems

Some Important Dates and People

**Introduction and Preliminaries**

Motivating Problems

Mathematical Background

Function Spaces

Mathematical Formulation of Problems

**The Simplest Problem in the Calculus of Variations **The Mathematical Formulation of the SPCV

The Fundamental Lemma of the Calculus of Variations

The First Necessary Condition for a Global Minimizer

Implications and Applications of the FLCV

**Necessary Conditions for Local Minima **Weak and Strong Local Minimizers

The Euler Necessary Condition - (I)

The Legendre Necessary Condition - (III)

Jacobi Necessary Condition - (IV)

Weierstrass Necessary Condition - (II)

Applying the Four Necessary Conditions

**Sufficient Conditions for the Simplest Problem **A Field of Extremals

The Hilbert Integral

Fundamental Sufficient Results

Summary for the Simplest Problem

**Extensions and Generalizations**

Properties of the First Variation

The Free Endpoint Problem

The Simplest Point to Curve Problem

Vector Formulations and Higher Order Problems

Problems with Constraints: Isoperimetric Problem

Problems with Constraints: Finite Constraints

An Introduction to Abstract Optimization Problems

**Applications **Solution of the Brachistochrone Problem

Classical Mechanics and Hamilton's Principle

A Finite Element Method for the Heat Equation

*Optimal Control ***Optimal Control Problems **

An Introduction to Optimal Control Problems

The Rocket Sled Problem

Problems in the Calculus of Variations

Time Optimal Control

**Simplest Problem in Optimal Control**

SPOC: Problem Formulation

The Fundamental Maximum Principle

Application of the Maximum Principle to Some Simple Problems

**Extensions of the Fundamental Maximum Principle **A Fixed-Time Optimal Control Problem

Application to Problems in the Calculus of Variations

Application to the Farmer’s Allocation Problem

Application to a Forced Oscillator Control Problem

Application to the Linear Quadratic Control Problem

The Maximum Principle for a Problem of Bolza

The Maximum Principle for Nonautonomous Systems

Application to the Nonautonomous LQ Control Problem

**Linear Control Systems **Introduction to Linear Control Systems

Linear Control Systems Arising from Nonlinear Problems

Linear Quadratic Optimal Control

The Riccati Differential Equation for a Problem of Bolza

Estimation and Observers

The Time Invariant Infinite Interval Problem

The Time Invariant Min-Max Controller

*Problems appear at the end of each chapter.*

**John Burns** is the Hatcher Professor of Mathematics, Interdisciplinary Center for Applied Mathematics at Virginia Polytechnic Institute and State University. He is a fellow of the IEEE and SIAM. His research interests include distributed parameter control; approximation, control, identification, and optimization of functional and partial differential equations; aero-elastic control systems; fluid/structural control systems; smart materials; optimal design; and sensitivity analysis.