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| In this talk, we derive and analyze variational integrators of higher order for the structure-preserving simulation and optimal control of mechanical systems. The construction is based on a space of polynomials together with Gauss and Lobatto quadrature rules to approximate the relevant integrals in the variational principle. The use of higher order schemes increases the accuracy of the discrete solution and thereby decrease the computational cost, while the preservation properties of the scheme are still preserved. The order of accuracy of the resulting variational integrators as well as stability properties are determined and demonstrated by numerical examples. Furthermore, by using theses integrators for the discretization of optimal control problems, we investigate the approximation order of the discrete adjoint equations resulting from the necessary optimality conditions of the discretized optimal control problem. |
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