Geodesic
On the proof of Pontryagin's maximum principle by means of needle variations
Fundamentalʹnaâ i prikladnaâ matematika, Tome 19 (2014) no. 5, pp. 49-73.

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We propose a proof of the maximum principle for the general Pontryagin type optimal control problem, based on packets of needle variations. The optimal control problem is first reduced to a family of smooth finite-dimensional problems, the arguments of which are the widths of the needles in each packet, then, for each of these problems, the standard Lagrange multipliers rule is applied, and finally, the obtained family of necessary conditions is “compressed” in one universal optimality condition by using the concept of centered family of compacta. 
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A. V. Dmitruk; N. P. Osmolovskii. On the proof of Pontryagin's maximum principle by means of needle variations. Fundamentalʹnaâ i prikladnaâ matematika, Tome 19 (2014) no. 5, pp. 49-73. http://geodesic.mathdoc.fr/item/FPM_2014_19_5_a2/

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