Structural stability without coercivity

We study structural stability and local uniqueness of bang-bang-singular extremals in a fixed-free Mayer problem. The problem can be seen as an optimal control problem where the dynamics is control-affine. We prove our results under the same assumptions on the nominal problem that permit us to prove that the extremal is a strict strong local optimiser: suitable regularity assumptions and coercivity of a restriction of the {em extended second variation}. We also assume that the condition that prevents the coercivity of the extended second variation is stable with respect to the parameter.