\documentclass{abstract}
\begin{document}

\title{Hamilton-Jacobi 
equations on networks as limits 
of singularly perturbed problems in optimal
control: dimension reduction}

\author{Yves Achdou}
 \institute{Univ. Paris Diderot,
% Sorbonne Paris Cit{\'e},
 Laboratoire Jacques-Louis Lions 
%,UMR 7598, UPMC, CNRS, F-75205 Paris, France
}
 \email{achdou@ljll.univ-paris-diderot.fr}
\coauthor{Nicoletta Tchou}
\mymaketitle

\section{Abstract}
  We consider a family of open star-shaped domains $\Omega^\epsilon$ of $I\!R^2$. Roughly speaking, 
$\Omega^\epsilon$ is made of a finite number of non intersecting semi-infinite strips of thickness $\epsilon$
and of a central region whose diameter is of the order of $\epsilon$, that may be called the junction. 
When the thickness  $\epsilon$ tends to $0$,
the domains $\Omega^\epsilon$ tend to a union of half-lines sharing an endpoint $O$. This set is termed {\sl network}.
    We study infinite horizon optimal control problems 
in which the state is constrained to remain in $\overline {\Omega^\epsilon}$. In the above mentioned strips the running cost may have a fast variation w.r.t. the transverse coordinate. 
We pass to the limit as the parameter $\epsilon$ tends to zero, and prove that the value function tends to the solution of a Hamilton-Jacobi equation on the network,
which may also be related to an optimal control problem. One difficulty is to find the transmission condition at the junction node $O$ in the limit problem. For passing to the limit, we use the method of the perturbed test-functions of Evans, which requires constructing suitable correctors. This is another difficulty since the domain is unbounded.


\paragraph{Acknowledgments}
The authors  were partially funded  by the ANR project ANR-12-BS01-0008-01.
The first author  was partially funded  by the ANR project ANR-12-MONU-0013.


\begin{thebibliography}{9} 

 \bibitem{AOT1}
Y. Achdou, S. Oudet and N.A. Tchou.
 \emph{Hamilton-Jacobi equations for optimal control on junctions and networks}
HAL : hal-00847210, 2013

\bibitem{AT-PertSing}
Y. Achdou and N.A. Tchou.
 \emph{Hamilton-Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction}
HAL :hal-00961015 ,2014

\end{thebibliography}

\end{document}
