Geodesic
Sufficient optimality conditions for linear differential inequalities with discontinuous trajectories
Izvestiya. Mathematics , Tome 41 (1993) no. 1, pp. 39-54.

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This article contains a formalization of the linear differential problem of optimal control with discontinuous trajectories in the form of a linear programming (LP) problem in a partially ordered space. This enables us to construct a dual LP problem and to obtain on this basis a sufficient criterion for optimality. The main idea in the construction of a dual pair of LP-problems consists in the introduction of an orientation in the space of piecewise continuous functions (left for the direct problem and right for the dual problem). The author thinks that the proposed criterion is close to necessary (for the class of functions under consideration), although he has not been able to prove it. The method of using the criterion for actually finding optimal trajectories is illustrated with an example; this method has actually been used by the author earlier to solve problems of an applied nature. 
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V. Z. Belen'kii. Sufficient optimality conditions for linear differential inequalities with discontinuous trajectories. Izvestiya. Mathematics , Tome 41 (1993) no. 1, pp. 39-54. http://geodesic.mathdoc.fr/item/IM2_1993_41_1_a1/

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