High order variational integrators in the optimal control of mechanical systems
Cédric M. Campos Sina Ober-Blöbaum Emmanuel Trélat
In recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class are variational integrators which are momentum preserving and symplectic. In this article, we develop two high order variational integrators which distinguish themselves in the dimension of the underling space of approximation and we investigate their application to finite-dimensional optimal control problems posed with mechanical systems. The convergence of state and control variables of the approximated problem is shown. Furthermore, by analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that, for these particular integrators, dualization and discretization commute.
keywords: high order Optimal control commutation property. direct methods geometric integration Runge-Kutta mechanical systems variational integrator
Classical field theories of first order and Lagrangian submanifolds of premultisymplectic manifolds
Cédric M. Campos Elisa Guzmán Juan Carlos Marrero
A description of classical field theories of first order in terms of Lagrangian submanifolds of premultisymplectic manifolds is presented. For this purpose, a Tulczyjew's triple associated with a fibration is discussed. The triple is adapted to the extended Hamiltonian formalism. Using this triple, we prove that Euler-Lagrange and Hamilton-De Donder-Weyl equations are the local equations defining Lagrangian submanifolds of a premultisymplectic manifold.
keywords: Tulczyjew's triple Field theory multisymplectic structure Lagrangian submanifold Hamilton-De Donder-Weyl equation. Euler-Lagrange equation

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