Part two - Optimal control theory

Summary

Much like the unidentified hero figure in classic folk tales who, after having won his position by virtue of his achievements, in the end reveals a hidden but distinguished parentage, so optimal control theory, recognized initially as an engineering subject, reveals, as it reaches maturity, a distinct relationship to classic forebears: the calculus of variations, differential geometry, and mechanics. This distinctive character of optimal control theory can be traced to the mathematical problems of the subject in the mid-1950s dealing with inequality constraints. Faced with the practical, time-optimal control problems of that period, mathematicians and engineers looked to the calculus of variations for answers, but soon discovered, through the papers of Bellman et al. (1956), LaSalle (1954), and Bushaw (1958) on the bang-bang controls, that the answers to their problems were outside the scope of the classic theory and would require different mathematical tools. That realization initiated a search for new necessary conditions for optimality suitable for control problems. That search, further intensified by the space program and the race to the moon, eventually led, in 1959, to the “maximum principle,” which answered the practical needs of that period. So strongly was the maximum principle linked to control problems involving bounds on the controls that its significance in a larger context – as an important extension of Weierstrass's excess function in the calculus of variations – went unnoticed for a long time after its discovery.

The treatment here of optimal control problems combines the rich theory of the calculus of variations with control-theory innovations. I believe, as does Young (1969), that a control-theory point of view is more natural for the calculus of variations.