A new Lagrangian approach to optimal control of second-order systems
Michael Konopik, Sigrid Leyendecker, Sofya Maslovskaya, Sina Ober-Blöbaum, Rodrigo T. Sato Martín de Almagro
TL;DR
This work develops a geometric framework for the optimal control of second-order mechanical systems by introducing a hyperregular, control-parameterized Lagrangian $\tilde{\mathcal{L}}^{\mathcal{E}}$ and its dual $\tilde{\mathcal{H}}^{\mathcal{E}}$, thereby reframing OCPs on an extended Tulczyjew triple that incorporates controls. It shows that this new Lagrangian–Hamiltonian formulation is equivalent to Pontryagin's maximum principle (PMP) and provides a canonical Tulczyjew-based extension with twisted sums $\alpha_\mathcal{Q}^{\mathcal{E}}$, $\beta_\mathcal{Q}^{\mathcal{E}}$ that unify force-controlled Lagrangian/Hamiltonian dynamics with OCPs. The paper also derives boundary-term interpretations via generating functions and establishes Noether-type conservation laws within this framework, linking symmetries of the OCP to conserved quantities in both the Lagrangian and PMP pictures. Overall, the approach yields a canonical, structure-preserving geometric formulation of second-order OCPs that facilitates discretization, reduction, and analysis for underactuated and force-controlled systems.
Abstract
In this work, we propose and study a new approach to formulate the optimal control problem of second-order differential equations, with a particular interest in those derived from force-controlled Lagrangian systems. The formulation results in a new hyperregular control Langrangian and, thus, a new control Hamiltonian whose equations of motion provide necessary optimality conditions. We compare this approach to Pontryagin's maximum principle (PMP) in this setting, providing geometric insight into their relation. This leads us to define an extended Tulczyjew's triple with controls. Moreover, we study the relationship between Noether symmetries of this new formulation and those of the PMP.