Introduction
The classical calculus of variations has been generalized.The maximum principle for optimal control, developed in the late 1950s by L.S. Pontryagin and his co-workers, applies to all calculus of variations problems. In such problems, optimal control gives equivalent results, as one would expect. The two approaches differ. However, and the optimal control approach sometimes affords insights into a problem that might be less readily apparent through the calculus of variations.
Optimal control also applies to problems for which the calculus of variations is not convenient, such as those involving constraints on the derivatives of functions sought. For instance,one can solve problems in which net investment or production rates are required to be nonnegative. While proof of the
maximum principle under full generality is beyond our scope, the now-familiar methods are used to generate some of the results of interest and to lend plausibility to others.