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

Sigrid Leyendecker; Sofya Maslovskaya; Sina Ober-Blöbaum; Rodrigo T. Sato Martín de Almagro; Flóra Orsolya Szemenyei

doi:10.3934/jcd.2024017

Article Contents

2024, Volume 11, Issue 3: 336-353. Doi: 10.3934/jcd.2024017

This issue Previous Article The Prytz connections Next Article Optimization via conformal Hamiltonian systems on manifolds

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

Sigrid Leyendecker^1,,
Sofya Maslovskaya^{2, †,},
Sina Ober-Blöbaum^2,,
Rodrigo T. Sato Martín de Almagro^{1, †,} and
Flóra Orsolya Szemenyei^{1, †, ,}

1.
Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Institute of Applied Dynamics (LTD), Immerwahrstrasse 1, 91058 Erlangen, Germany
2.
Universität Paderborn (UPB), Numerical Mathematics and Control (NMC), Warburger Straße 100, 33098 Paderborn, Germany

^*Corresponding author: Flóra Orsolya Szemenyei
^*Corresponding author: Flóra Orsolya Szemenyei

^†First authors: Sofya Maslovskaya, Rodrigo T. Sato Martín de Almagro and Flóra Orsolya Szemenyei

Received: July 2023

Revised: February 2024

Early access: March 2024

Published: July 2024

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Abstract

In this work, we consider optimal control problems for mechanical systems 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. 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.

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Mathematics Subject Classification: Primary: 65K10, 49M25; Secondary: 65K15.

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Figure 1. Schematic representation of the connection between Lagrangian and Hamiltonian mechanics. Here, $ (q(t), \dot{q}(t)) $ stands for a solution of the Euler-Lagrange equations corresponding to the Lagrangian $ L $, and $ (q(t), p(t)) $ for a solution of Hamilton's equations corresponding to the Hamiltonian $ H $. The Legendre transformation $ \mathbb{F}L $ connects the Lagrangian and Hamiltonian sides of mechanics and therefore the Euler-Lagrange equations and Hamilton's equations

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