Variational integrators for a new Lagrangian approach to control affine systems with a quadratic Lagrange term

TL;DR

This paper develops a novel Lagrangian-based discretisation method for control affine systems with quadratic costs, deriving exact and approximate optimality conditions and applying them to orbital transfer problems.

Contribution

It introduces exact and semi-discrete formulations for a new Lagrangian approach, along with low-order schemes and a Noether's theorem extension for optimal control.

Findings

Derived discrete necessary conditions for optimality.

Proposed low-order integration schemes for approximate solutions.

Successfully applied methods to a low-thrust orbital transfer problem.

Abstract

In this work, we analyse the discretisation of a recently proposed new Lagrangian approach to optimal control problems of affine-controlled second-order differential equations with cost functions quadratic in the controls. We propose exact discrete and semi-discrete versions of the problem, providing new tools to develop numerical methods. Discrete necessary conditions for optimality are derived and their equivalence with the continuous version is proven. A family of low-order integration schemes is devised to find approximate optimality conditions, and used to solve a low-thrust orbital transfer problem. Non-trivial equivalent standard direct methods are constructed. Noether's theorem for the new Lagrangian approach is investigated in the exact and approximate cases.