Geodesic
Duality relations for the convex optinal control problem
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 23 (1983) no. 1, pp. 39-50

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Extremum conditions in the form of a maximum principle for the optimal control problem are derived from extrenum conditions for the finite-difference analogues. On the basis of the optimal values of the direct and dual variables for a finite-difference analogue, upper and lower bounds are evaluated for the optimum value of the functional of the continuous problem, and these bounds are shown to tend to the optimum functional in question. 
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     author = {A. A. Levikov},
     title = {Duality relations for the convex optinal control problem},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {39--50},
     publisher = {mathdoc},
     volume = {23},
     number = {1},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_1983_23_1_a4/}
}
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A. A. Levikov. Duality relations for the convex optinal control problem. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 23 (1983) no. 1, pp. 39-50. http://geodesic.mathdoc.fr/item/ZVMMF_1983_23_1_a4/