Structural stability for bang--singular--bang extremals in the minimum time problem

We study the structural stability of a bang-singular-bang extremal in the minimum time problem between fixed points. The dynamics is single-input and control-affine with bounded control. On the nominal problem ($r = 0$), we assume the coercivity of a suitable second variation along the singular arc and regularity both of the bang arcs and of the junction points, thus obtaining the strict strong local optimality for the given bang-singular-bang extremal trajectory. Moreover, as in the classically studied regular cases, we assume a suitable controllability property, which grants the uniqueness of the adjoint covector. Under these assumptions we prove that, for any sufficiently small $r$, there is a bang-singular-bang extremal trajectory which is a strict strong local optimiser for the $r$-problem. A uniqueness result in a neighbourhood of the graph of the nominal extremal pair is also obtained. The results are proven via the Hamiltonian approach to optimal control and by taking advantage of the implicit function theorem, so that a sensitivity analysis could also be carried out.