Introduction to the Calculus of Variations and Control with Modern Applications

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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.

Table of Contents

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

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.

Author Bio(s)

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.