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\title{On the Optimization of Riemann-Stieltjes-Control-Systems with Application in Vehicle Dynamics}
\author{Johannes Michael}
\institute{University of the Federal Armed Forces Munich}
\email{johannes.michael@unibw.de}
\coauthor{Matthias Gerdts} % comment out if not needed

\mymaketitle

\section{Abstract}
Optimizing discontinuous systems is a field of research with many unsolved problems. In this work, we focus on a method to model dynamic multibody control systems subject to state dependent frictional impacts using Riemann-Stieltjes-Integrals. As the impulsive character of the system is based on the state only, the controls exclusively enter the continuous Riemannian part of the dynamics. The system equations are combined with a set of impact conditions, wich are introduced to calculate the jumps of generalized velocities at contact using a formulation with energetically consistent generalized coefficients-of-restitution. These impact conditions are a reformulation of a complementarity problem with additional constraints on the post-impact velocities. Based on this model of differential equations with state discontinuities, a gradient based optimization routine is presented making use of the bounded variation assumption in the velocity components. The velocity jumps are computed by evaluatingthe frictional model as a minor optimization problem for the coefficients-of-restitution in tangential direction. The theoretical method has shown to be applicable for different optimization problems with continuous state dependent cost functionals. The method is illustrated in a numerical example, where comfort and handling characteristics for vehicle models are optimized due to road irregularities. Last, we draw a comparison between results derived by this method and a commercial library for tyre-road interaction forces.
%\cite{pfeiffer2008multibody} \cite{schmaedeke1965} \cite{Tortorella1990} \cite{bowling2009energetically} \cite{brach1998formulation} \cite{stewart2009}

\paragraph{Acknowledgments}
This work was partially funded by the German Federal Ministry of Education
and Research (BMBF), grant no. 05M10WNA.

\begin{thebibliography}{9}

\bibitem{bowling2009energetically}
Alan Bowling, Daniel Montrallo Flickinger, and Sean Harmeyer.
\newblock Energetically consistent simulation of simultaneous impacts and contacs in multibody systems with friction
\newblock {\em Multibody System Dynamics}, 22(1):27--45, 2009.

\bibitem{brach1998formulation}
Raymond~M. Brach.
\newblock Formulation of rigid body impact problems using generalized coefficients.
\newblock {\em International journal of engineering science}, 36(1):61--71,
  1998.

\bibitem{pfeiffer2008multibody}
F.~Pfeiffer and C.~Glocker.
\newblock {\em Multibody Dynamics with Unilateral Contacts}.
\newblock Wiley Series in Nonlinear Science. Wiley, 2008.

\bibitem{schmaedeke1965}
W.~W. Schmaedeke.
\newblock {Optimal Control Theory for Nonlinear Vector Differential Equations Containing Measures}.
\newblock {\em Journal of The Society for Industrial and Applied Mathematics,
  Series A: Control}, 3, 1965.

\bibitem{stewart2009}
David E. Stewart and Mihai Anitescu.
\newblock Optimal control of systems with discontinuous differential equations.
\newblock {\em Numerische Mathematik}, 114(4):653--695, 2010.

\bibitem{Tortorella1990}
Michael Tortorella.
\newblock Closed newton-cotes quadrature rules for stieltjes integrals and numerical convolution of life distributions.
\newblock {\em SIAM J. Sci. Stat. Comput.}, 11(4):732--748, July 1990.

\end{thebibliography}

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