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Remarks on Optimal Control I: The Standard Sufficiency Theory for the Least Time Problem

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Technical Report

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The sufficiency theory treated in the report concerns a least time problem of optimal control in which the suspected solutions, which obey a strengthened form of the Pontryagin maximum principle, cover a certain set E in n-space in a particular manner that we describe. The set E need not be a domain, and the covering need not be simple. In spite of this, we are able to develop a theory similar to that of Caratheodory in the classical Calculus of Variations. This theory is, however, now valid in the large, and in circumstances which differ rather radically from those which occur in the classical case. The main tools are a greatly strengthened theorem of Malus on the one hand, and the use, on the other hand, of a new and more powerful Hilbert independence integral, whose integrand is now, in effect, many-valued. The very general situations, to which the theory is applicable, require moreover, many new concepts and definitions, which are, in part, of a topological nature, and which make it possible to avoid restrictions to one-to-one maps with non-singular Jacobians, restrictions which are normally made in the Calculus of Variations, but which would here be quite out of place.

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