Geodesic
Controllability and second-order necessary conditions for optimality
Sbornik. Mathematics, Tome 210 (2019) no. 1, pp. 1-23

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Sufficient conditions for the local controllability of a control system of ordinary differential equations are presented. These conditions are meaningful in the case where the linear approximation of the system is not completely controllable. Second-order necessary conditions for optimality for the problem of optimal control are obtained as a corollary. Bibliography: 13 titles.
Keywords: controllability, optimal control, second-order conditions. 
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E. R. Avakov; G. G. Magaril-Il'yaev. Controllability and second-order necessary conditions for optimality. Sbornik. Mathematics, Tome 210 (2019) no. 1, pp. 1-23. http://geodesic.mathdoc.fr/item/SM_2019_210_1_a0/