Geodesic
Generalized Needles and Second-Order Conditions in Optimal Control
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal control and differential equations, Tome 304 (2019), pp. 15-31

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Using generalized needle variations, we derive second-order necessary optimality conditions for a strong minimum in an optimal control problem with endpoint constraints of general form. 
@article{TM_2019_304_a1,
     author = {E. R. Avakov and G. G. Magaril-Il'yaev},
     title = {Generalized {Needles} and {Second-Order} {Conditions} in {Optimal} {Control}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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     volume = {304},
     year = {2019},
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     url = {http://geodesic.mathdoc.fr/item/TM_2019_304_a1/}
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E. R. Avakov; G. G. Magaril-Il'yaev. Generalized Needles and Second-Order Conditions in Optimal Control. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal control and differential equations, Tome 304 (2019), pp. 15-31. http://geodesic.mathdoc.fr/item/TM_2019_304_a1/