Calculus of Variations: Mechanics, Control and Other Applications

by Charles R. MacCluer

Overview

The first truly up-to-date treatment of the calculus of variations, this text is also the first to offer a simple introduction to such key concepts as optimal control and linear-quadratic control design. Suitable for junior/senior–level students of math, science, and engineering, this volume also serves as a useful reference for engineers, chemists, and forest/environmental managers. Its broad perspective features numerous exercises, hints, outlines, and comments, plus several appendixes, including a practical discussion of MATLAB.
Students will appreciate the text's reader-friendly style, which features gradual advancements in difficulty and starts by developing technique rather than focusing on technical details. The examples and exercises offer many citations of engineering-based applications, and the exercises range from elementary to graduate-level projects, including longer projects and those related to classic papers.

Product Details

ISBN-13:	9780486278308
Publisher:	Dover Publications
Publication date:	04/22/2013
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	272
File size:	12 MB
Note:	This product may take a few minutes to download.

Table of Contents

PrefaceAcknowledgments1. Preliminaries1.1 Directional Derivatives and Gradients1.2 Calculus Rules1.3 Contour Surfaces and Sublevel Sets1.4 Lagrange Multipliers1.5 ConvexityExercises2. Optimization2.1 Mathematical Programming2.2 Linear Programming2.3 Statistical Problems2.4 Variational ProblemsExercises3. Formulating Variational Problems3.1 Shortest Distance between Two Points (CVP 1)3.2 Graph with Least Surface of Revolution (CVP 2)3.3 The Catenary (CVP 3)3.4 The Brachistochrone (CVP 4)3.5 Cruise-Climb (CVP 5)3.6 Shapes of Minimum Resistance (CVP 6)3.7 Hamilton's Principle3.8 Isoperimetric ProblemsExercises4. The Euler-Lagrange Equation4.1 One Degree of Freedom4.2 Two Special Cases: No y, No x4.3 Multiple Degrees of Freedom4.4 The Hamiltonian4.5 A Closer LookExercises5. Constrained Problems5.1 Dido's Problem5.2 Statement of the Problem5.3 The Inverse Function Theorem5.4 The Euler-Lagrange Equation for Constrained Problems5.5 Example Applications5.6 Multiple Degrees of Freedom5.7 Nonintegral Constraints5.8 Hamilton's Principle with ConstraintsExercises6. Extremal Surfaces6.1 A Soap Film (CVP 15)6.2 Stable Flows (CVP 17)6.3 Schrodinger's Equation (CVP 18)6.4 Eigenvalue Problems6.5 Rayleigh-Ritz NumericsExercises7. Optimal Control7.1 A Rolling Cart (OCP 1)7.2 General Formulation 7.3 Reinvestments (OCP 2)7.4 Average Voltage (OCP 3)7.5 A Time-Optimal Problem (OCP 4)7.6 The Bang-Bang Principle7.7 The Maximum Principle7.8 Example ApplicationsExercises8. The LQ Problem8.1 Problem Statement8.2 State Feedback8.3 Stability8.4 The LQR Problem8.5 A Tracking ServoExercises9. Weak Sufficiency9.1 Weak versus Strong Extrema9.2 First and Second Variations9.3 In Application9.4 The Integrand9.5 Weak Local SufficiencyExercises10. Strong Sufficiency10.1 The Goal10.2 Flows10.3 Flows of the Euler-Lagrange Equation10.4 The E-Function and Strong Sufficiency10.5 The Existence of FlowsExercises11. Corner Points11.1 Corners and Extremals11.2 First Erdmann Corner Condition11.3 The Figurative11.4 Second Erdmann Corner ConditionExercisesAppendix A. The Inverse Function TheoremAppendix B. Picard's TheoremAppendix C. The Divergence TheoremAppendix D. A MATLAB CookbookReferencesIndex