Variational methods in problems of control and programming

by Leonard David Berkovitz

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A study that shows a fairly general control problem, or programming problem, with constraints can be reduced to a special type of classical Bolza problem in the calculus of variations. Necessary conditions from the Bolza problem are thanslated into necessary conditions for optimal control. It is seen from these conditions that Pontryagin's maximum principle is a translation of the usual Weierstrass consition, and is applicable to a wider class of problems than that considered by Pontryagin. The differentiability and continuity properties of the valu of the control are established under reasonable hypotheses on the synthesis, and it is shown that the value satisfies the Hamilton-Jacobi equation. As a consequence, a rigorous proof of a functional equation of Bellman is obtained that is valid for much wider class of problems than heretofore considered. A sufficiency theorem for synthesis

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