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This paper presents the variational approach to some optimization problems: Mayer's problem with or without constraints on the final point, local controllability of a trajectory, time-optimal problems.
It is shown that the Lagrange's equations for a Lagrangian system on a Lie algebroid are obtained as the equations for the critical points of the action functional defined on a Banach manifold of curves. The theory of Lagrangian reduction and the relation with the method of Lagrange multipliers are also studied.
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