Publisher Summary This chapter discusses the problem of minimizing a quadratic functional using the methods of the calculus of variations. To treat this problem rigorously in a simple fashion, the chapter employs a formal method to obtain the Euler equation, a fundamental necessary condition. Then, it shows by means of a simple direct calculation that this equation has a unique solution, and that this solution provides the desired minimum value. In the calculus of variations, emphasis is customarily placed upon establishing the existence of a solution over some T-interval, however small. In control theory, it is natural to examine the nature of the solution for all T > 0. In a number of important control processes, the criterion for efficiency is a mixture of the history of the process and the final state of the system. The chapter also discusses terminal control.