Geodesic
The first variation and Pontryagin's maximum principle in optimal control for partial differential equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 6, pp. 998-1020

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A modification of the classical needle variation, namely, the so-called two-parameter variation of controls is proposed. The first variation of a functional is understood as a repeated limit. It is shown that the modified needle variation can be effectively used to derive necessary optimality conditions for a rather wide class of optimal control problems involving partial differential equations with weak solutions. Specifically, the two-parameter variation is used to obtain necessary optimality conditions in the form of a maximum principle for the optimal control of divergent hyperbolic equations. 
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     title = {The first variation and {Pontryagin's} maximum principle in optimal control for partial differential equations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {998--1020},
     publisher = {mathdoc},
     volume = {49},
     number = {6},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_6_a5/}
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M. I. Sumin. The first variation and Pontryagin's maximum principle in optimal control for partial differential equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 6, pp. 998-1020. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_6_a5/