Geodesic
Controllability and Second-Order Necessary Conditions for Local Infimum Trajectories in Optimal Control
Informatics and Automation, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 7-30.

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We introduce the concept of local controllability of a dynamical system and present sufficient conditions for such controllability. As an immediate consequence, we prove necessary second-order conditions for a local infimum trajectory in an optimal control problem. These conditions strengthen the well-known second-order optimality conditions and extend them to more general classes of problems. 
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E. R. Avakov; G. G. Magaril-Il'yaev. Controllability and Second-Order Necessary Conditions for Local Infimum Trajectories in Optimal Control. Informatics and Automation, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 7-30. http://geodesic.mathdoc.fr/item/TRSPY_2023_321_a0/

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