Minimum-time strong optimality of a singular arc: the multi-input non involutive case

TL;DR

This paper proves that a second variation condition guarantees the strong local optimality of singular arcs in minimum-time control problems for multi-input systems with non-involutive distributions, extending classical results.

Contribution

It introduces a Hamiltonian-based second variation criterion for strong optimality of singular arcs in multi-input control-affine systems without control bounds.

Findings

Second variation coercivity ensures strong local optimality.

Application to a generalized Dubins problem.

Extension to non-involutive distribution cases.

Abstract

We consider the minimum-time problem for a multi-input control-affine system, where we assume that the controlled vector fields generate a non-involutive distribution of constant dimension, and where we do not assume a-priori bounds for the controls. We use Hamiltonian methods to prove that the coercivity of a suitable second variation associated to a Pontryagin singular arc is sufficient to prove its strong-local optimality. We provide an application of the result to a generalization of Dubins problem.