Geodesic
Schauder's fixed point theorem and Pontryagin maximum principle
Izvestiya. Mathematics , Tome 88 (2024) no. 6, pp. 1013-1031

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We prove the Pontryagin maximum principle for a general optimal control problem. The main ingredient of the proof is the abstract lemma on an inverse function, which is proved via the Schauder fixed-point theorem. Under this approach, the proof of the Pontryagin maximum principle is quite short and transparent.
Keywords: the Pontryagin maximum principle, the Schauder fixed-point theorem, inverse function lemma. 
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     title = {Schauder's fixed point theorem and {Pontryagin} maximum principle},
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E. R. Avakov; G. G. Magaril-Il'yaev. Schauder's fixed point theorem and Pontryagin maximum principle. Izvestiya. Mathematics , Tome 88 (2024) no. 6, pp. 1013-1031. http://geodesic.mathdoc.fr/item/IM2_2024_88_6_a0/