THE MAXIMUM PRINCIPLE, BELLMAN EQUATION, AND CARATHEODORY WORK

One of the most important and deep results in optimal control theory i s the maximum principle attributed to Hestenes (1950) and in particula r to Boltyanskii, Gamkrelidze, and Pontryagin (1956). Another prominen t result is known as the Bellman equation, which is associated with Is aacs' and Bellman's work (later than 1951). However, precursors of bot h the maximum principle and the Bellman equation can already be found in Caratheodory's book of 1935 (Ref. la), the first even in his earlie r work of 1926 which is given in Ref. 2. This is not a widely acknowle dged fact. The present tutorial paper traces Caratheodory's approach t o the calculus of variations, once called the ''royal road in the calc ulus of variations,'' and shows that famous results in optimal control theory, including the maximum principle and the Bellman equation, are consequences of Caratheodory's earlier results.