SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Contributed Talk

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On higher order variational schemes for numerical optimal control

S. Ober-Blöbaum

Direct methods for optimal control problems rely on a discretization of the underlying dynamical system. The method DMOC [2] is based on variational integrators [1] such that geometric properties of the system are preserved in the discrete solution. To ensure moderate computational costs, typically first or second order integrators are used. However, many applications demand more accurate discretization schemes. We derive and analyze higher order variational schemes for the structure-preserving simulation of mechanical systems. This leads to a higher accuracy of the discrete solution and less computational cost, while geometric properties are still preserved. The order of accuracy and stability properties of the integrators are analytically and numerically determined [3]. Furthermore, we investigate the approximation order of the discrete adjoint equations resulting from the necessary optimality conditions of the discretized optimal control problem.

[1] J. E. Marsden and M. West. Discrete mechanics and variational integrators, Acta Numerica, 10 (2001) 357-514.
[2] S. Ober-Blöbaum, O. Junge and J. E. Marsden. Discrete mechanics and optimal control: an analysis, Control, Optimisation and Calculus of Variations, 17(2) (2011) 322-352.
[3] S. Ober-Blöbaum and N. Saake. Construction and analysis of higher order Galerkin variational integrators, Submitted, 2013.

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