This book presents a rigorous introduction to variational calculus and optimal control, with a focus on function spaces of piecewise continuous and smooth functions. It shows how the calculus of variations can analyze optimization problems in functional analysis, optimal control, mechanics, and partial differential equations. It also illustrates how variational calculus provides the mathematical framework for a wide range of problems in engineering such as finite element analysis. The book provides readers with a sound understanding of the difference between necessary and sufficient conditions and when to use them.
Part I Calculus of Variations: Historical Notes on the Calculus of Variations. Introduction and Preliminaries. The Simplest Problem in the Calculus of Variations. Necessary Conditions for Local Minima. Sufficient Conditions for the Simplest Problem. Summary for the Simplest Problem. Extensions and Generalizations. A Review of General Necessary Conditions. Applications. Part II Optimal Control: Optimal Control Problems. Simplest Problem in Optimal Control. Extensions of the Fundamental Maximum Principle. Linear Control Systems.