Geodesic
Symplectic geometry and conditions necessary conditions for optimality
Sbornik. Mathematics, Tome 72 (1992) no. 1, pp. 29-45 
A. A. Agrachev; R. V. Gamkrelidze. Symplectic geometry and conditions necessary conditions for optimality. Sbornik. Mathematics, Tome 72 (1992) no. 1, pp. 29-45. http://geodesic.mathdoc.fr/item/SM_1992_72_1_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

With the help of a symplectic technique the concept of a field of extremals in the classical calculus of variations is generalized to optimal control problems. This enables us to get new optimality conditions that are equally suitable for regular, bang-bang, and singular extremals. Special attention is given to systems of the form $\dot x=f_0(x)+uf_1(x)$ with a scalar control. New pointwise conditions for optimality and sufficient conditions for local controllability are obtained as a consequence of the general theory.

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