Singular extremals in L^1 optimal control problems: sufficient optimality conditions

TL;DR

This paper develops sufficient conditions for the strong local optimality of extremals in L^1 optimal control problems with control-affine dynamics, using Hamiltonian methods and providing explicit formulas for the second variation.

Contribution

It introduces new sufficient optimality conditions for singular extremals in L^1 control problems and derives an explicit invariant formula for the second variation along singular arcs.

Findings

Established sufficient conditions for local optimality of extremals.

Derived an explicit invariant formula for the second variation.

Validated conditions using Hamiltonian methods.

Abstract

In this paper we are concerned with generalised L 1-minimisation problems, i.e. Bolza problems involving the absolute value of the control with a control-affine dynamics. We establish sufficient conditions for the strong local optimality of extremals given by the concatenation of bang, singular and inactive (zero) arcs. The sufficiency of such conditions is proven by means of Hamiltonian methods. As a byproduct of the result, we provide an explicit invariant formula for the second variation along the singular arc.