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

TL;DR

This paper establishes sufficient conditions under which a specific bang-bang-singular extremal trajectory is a strong local minimizer in a Mayer optimal control problem with fixed-free endpoint constraints, using a Hamiltonian approach.

Contribution

It introduces new second order conditions for strong local optimality of bang-bang-singular extremals in fixed-free endpoint Mayer problems.

Findings

Provides sufficient conditions for strong local optimality.

Connects Hamiltonian approach with second order conditions.

Includes two illustrative examples.

Abstract

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