A new Lagrangian approach to control affine systems with a quadratic Lagrange term

TL;DR

This paper introduces a novel Lagrangian method for optimal control of mechanical systems with affine controls, enabling geometric insights and symplectic discretisation through Euler-Lagrange equations.

Contribution

It develops a new Lagrangian framework for control affine systems with quadratic costs, providing an alternative to Pontryagin's principle and facilitating geometric numerical methods.

Findings

Derivation of Euler-Lagrange equations for control affine systems

Enabling symplectic discretisation via variational integrators

Provides a geometric perspective on optimal control problems

Abstract

In this work, we consider optimal control problems for mechanical systems on vector spaces with fixed initial and free final state and a quadratic Lagrange term. Specifically, the dynamics is described by a second order ODE containing an affine control term and we allow linear coordinate changes in the configuration space. Classically, Pontryagin's maximum principle gives necessary optimality conditions for the optimal control problem. For smooth problems, alternatively, a variational approach based on an augmented objective can be followed. Here, we propose a new Lagrangian approach leading to equivalent necessary optimality conditions in the form of Euler-Lagrange equations. Thus, the differential geometric structure (similar to classical Lagrangian dynamics) can be exploited in the framework of optimal control problems. In particular, the formulation enables the symplectic discretisation of the optimal control problem via variational integrators in a straightforward way.