Quadratic conditions for a Pontryagin minimum in an optimum control problem linear in the control. I: A decoding theorem

The general optimum control problem considered here is linear in the control and without constraints on the control. Quadratic (i.e., second-order) necessary and sufficient conditions are given for the problem to have a minimum in the class of variations bounded in modulus by an arbitrary constant and having small integral. These conditions are stronger than the previously known conditions for a weak minimum, and, like the latter conditions, constitute an adjoining pair, i.e., the sufficient condition differs from the necessary condition only in the strengthening of an inequality.
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