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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.