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

TL;DR

This paper establishes the structural stability of bang-singular-bang extremals in minimum time control problems, showing that under certain conditions, these extremals remain locally optimal when small perturbations are introduced.

Contribution

The paper proves the stability and local optimality of bang-singular-bang extremals under perturbations, extending classical results to perturbed problems using Hamiltonian methods.

Findings

Existence of a stable extremal for small perturbations

Uniqueness of the extremal in a neighborhood

Local optimality preserved under perturbations

Abstract

In this paper 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. 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.