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In this paper we give sufficient conditions for a Pontryagin extremaltrajectory, consisting of two bang arcs followed by a singular one, to be astrong local minimizer for a Mayer problem. The problem is defined on amanifold $M$ and the end-points constraints are of fixed-free type. We use aHamiltonian approach and its connection with the second order conditions in theform of an accessory problem on the tangent space to $M$ at the final point ofthe trajectory. Two examples are proposed.

Strong local optimality for a bang-bang-singular extremal: the fixed-free case