Geodesic
Quadratic conditions for a~Pontryagin minimum in an optimum control problem linear in the control.~I: A~decoding theorem
Izvestiya. Mathematics , Tome 28 (1987) no. 2, pp. 275-303

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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. Bibliography: 17 titles. 
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     author = {A. V. Dmitruk},
     title = {Quadratic conditions for {a~Pontryagin} minimum in an optimum control problem linear in the {control.~I:} {A~decoding} theorem},
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A. V. Dmitruk. Quadratic conditions for a~Pontryagin minimum in an optimum control problem linear in the control.~I: A~decoding theorem. Izvestiya. Mathematics , Tome 28 (1987) no. 2, pp. 275-303. http://geodesic.mathdoc.fr/item/IM2_1987_28_2_a2/