Strong Local Optimality for a Bang-Bang-Singular Extremal: The Fixed-Free Case

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.

Constrained bang-bang-singular extremals

“…In proving our result we demonstrate that the fixed-free case treated in [1] is a case study in the Hamiltonian approach. Indeed, in considering the fixed-free case, the authors prove that certain regularity assumptions on the extremal and the coercivity of a suitable second variation allow to lift any neighboring admissible trajectory to the cotangent bundle.…”

“…The trajectory ξ of the system is called a state extremal of problem (1) while the couple λ(t) := µ(t), ξ(t) is called an extremal of problem (1). We denote the terminal points and the switching points of the reference extremal as…”

“…For the sake of completeness we state the regularity assumptions which are the same as in [1]. We refer to that paper for more details.…”

“…Remark 2.1: Thanks to (3), see e.g. [1], the singular control is smooth on ( τ 2 , T ). Moreover (3) implies that there exists a neighborhood O s of the range of the singular arc λ([ τ 2 , T ]) in T * R n such that the sets…”

“…The aim of this paper is to give an extension of the results obtained in [1]. In there the authors proved sufficient conditions for the strong local optimality of a Pontryagin extremal in a Mayer problem with fixed initial point and free final one and where the control set is given by a compact convex polyhedron.…”