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

Sigrid Leyendecker, Sofya Maslovskaya, Sina Ober-Blobaum, Rodrigo T. Sato Martin de Almagro, Flora Orsolya Szemenyei

TL;DR

The paper tackles optimal control of mechanical systems with quadratic Lagrange terms and control-affine dynamics by introducing a new regular Lagrangian on an extended state–adjoint space, yielding Euler-Lagrange equations that are equivalent to the classical PMP conditions. This approach enables direct use of Lagrangian and geometric methods, including Tulczyjew-style structures and a straightforward Legendre transform to a Hamiltonian formulation, with an eye toward variational discretisation via symplectic integrators. It also develops Noether-type conserved quantities arising from symmetries, illustrated through an SO(3) example, and demonstrates a concrete low-thrust orbital transfer as a mechanical application. The framework promises robust numerical integration through variational schemes and offers a path to generalisations to broader second-order dynamical constraints, regularity assumptions, and multisymplectic formulations, with potential impact on structure-preserving numerical optimal control.

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.

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

TL;DR

The paper tackles optimal control of mechanical systems with quadratic Lagrange terms and control-affine dynamics by introducing a new regular Lagrangian on an extended state–adjoint space, yielding Euler-Lagrange equations that are equivalent to the classical PMP conditions. This approach enables direct use of Lagrangian and geometric methods, including Tulczyjew-style structures and a straightforward Legendre transform to a Hamiltonian formulation, with an eye toward variational discretisation via symplectic integrators. It also develops Noether-type conserved quantities arising from symmetries, illustrated through an SO(3) example, and demonstrates a concrete low-thrust orbital transfer as a mechanical application. The framework promises robust numerical integration through variational schemes and offers a path to generalisations to broader second-order dynamical constraints, regularity assumptions, and multisymplectic formulations, with potential impact on structure-preserving numerical optimal control.

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.
Paper Structure (14 sections, 5 theorems, 74 equations, 1 figure)

This paper contains 14 sections, 5 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries, notations
  3. Lagrangian and Hamiltonian mechanics
  4. Optimal control problem
  5. Necessary optimality conditions
  6. Construction of a new Lagrangian for a control affine system with a specific objective
  7. Pontryagin's maximum principle for the control affine system
  8. Augmented objective approach
  9. New Lagrangian and Euler-Lagrange equations
  10. New Hamiltonian system
  11. Geometric setting
  12. Noether's theorem, conserved quantities
  13. A mechanical example - Low thrust orbital transfer
  14. Outlook on future work

Key Result

Theorem 3.2

Considering the optimal control problem eq:eqample.OCP. the following equations hold Moreover, for the final time one has

Figures (1)

  • 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.

Theorems & Definitions (16)

  • Remark 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Remark 3.5
  • Remark 3.6
  • Theorem 3.7
  • proof
  • ...and 6 more