ABSTRACT
This chapter provides a brief introduction to optimal control theory. It focuses on examples and problems as opposed to formal theory. It is difficult to obtain analytic solutions for even those simple problems which have closed form solutions. Even more difficult is to obtain formal proofs of methods such as the Pontryagin Principle. In a sense, this subject can be thought of as a modern form of constrained calculus of variations beginning with the work of the Pontryagin group in 1962. It initially appeared that the earlier theory of the calculus of variations was a subset of optimal control theory. This extension has many advantages, some of which are that it implied many necessary and sufficient conditions not easily obtained in the optimal control setting. In addition, Hestenes was immediately able to combine these areas and achieve a very general theory of optimal control.
