Geodesic
Necessary optimality conditions for a smooth discrete optimal control problem with vector-valued objective function
Trudy Instituta matematiki, Tome 17 (2009) no. 1, pp. 27-40

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The paper deals with a nonlinear discrete optimal control problem with a vector-valued objective function of terminal type. For admissible controls which satisfy the additional Lyusternik-type regularity condition, first and second necessary optimality conditions extending such classical optimality conditions as the Euler condition and the nonnegativity of second variation are proved. 
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     title = {Necessary optimality conditions for a smooth discrete optimal control problem with vector-valued objective function},
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V. V. Gorokhovik; S. Ya. Gorokhovik; B. Marinković. Necessary optimality conditions for a smooth discrete optimal control problem with vector-valued objective function. Trudy Instituta matematiki, Tome 17 (2009) no. 1, pp. 27-40. http://geodesic.mathdoc.fr/item/TIMB_2009_17_1_a2/