@article{53101,
  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.}},
  author       = {{Leyendecker, Sigrid and Maslovskaya, Sofya and Ober-Blöbaum, Sina and Almagro, Rodrigo T. Sato Martín de and Szemenyei, Flóra Orsolya}},
  issn         = {{2158-2491}},
  journal      = {{Journal of Computational Dynamics}},
  keywords     = {{Optimal control problem, Lagrangian system, Hamiltonian system, Variations, Pontryagin's maximum principle.}},
  pages        = {{0--0}},
  publisher    = {{American Institute of Mathematical Sciences (AIMS)}},
  title        = {{{A new Lagrangian approach to control affine systems with a quadratic Lagrange term}}},
  doi          = {{10.3934/jcd.2024017}},
  year         = {{2024}},
}

@article{19945,
  abstract     = {{Many PDEs (Burgers' equation, KdV, Camassa-Holm, Euler's fluid equations, …) can be formulated as infinite-dimensional Lie-Poisson systems. These are Hamiltonian systems on manifolds equipped with Poisson brackets. The Poisson structure is connected to conservation properties and other geometric features of solutions to the PDE and, therefore, of great interest for numerical integration. For the example of Burgers' equations and related PDEs we use Clebsch variables to lift the original system to a collective Hamiltonian system on a symplectic manifold whose structure is related to the original Lie-Poisson structure. On the collective Hamiltonian system a symplectic integrator can be applied. Our numerical examples show excellent conservation properties and indicate that the disadvantage of an increased phase-space dimension can be outweighed by the advantage of symplectic integration.}},
  author       = {{McLachlan, Robert I and Offen, Christian and Tapley, Benjamin K}},
  issn         = {{2158-2505}},
  journal      = {{Journal of Computational Dynamics}},
  number       = {{1}},
  pages        = {{111--130}},
  publisher    = {{American Institute of Mathematical Sciences (AIMS)}},
  title        = {{{Symplectic integration of PDEs using Clebsch variables}}},
  doi          = {{10.3934/jcd.2019005}},
  volume       = {{6}},
  year         = {{2019}},
}