\documentclass{abstract}
\begin{document}

\title{Regularization of chattering phenomena via bounded variation controls}
\author{Roberta Ghezzi}
\institute{Institut de Math\'ematiques de Bourgogne, 
 Dijon France}
\email{roberta.ghezzi@u-bourgogne.fr}
\coauthor{M.~Caponigro, B.~Piccoli, \and E.~Tr\'elat}  %

\mymaketitle

\section{Abstract}
In optimal control there may be bad oscillatory phenomena. For instance, this is the case for Fuller's phenomenon \cite{fuller} where the optimal control is bang-bang with an infinite number of switchings in  finite time. 
In problems with state constraints of high order, there are examples in which the optimal  trajectory touches the boundary of the constraint at an infinite number of points over a finite time interval \cite{robbins}. Likewise, 
in the framework of hybrid systems, there are optimization problems where the discrete part of the dynamics switches infinitely many times in finite time \cite{zeno-numpb} (and this is known as Zeno's phenomenon.) 

In \cite{paper} we provide a technique to regularize chattering controls. Given a general optimal control problem, we define a perturbed problem introducing a term of total variation in the cost and prove that the solution to the perturbed problem is quasi-optimal for the reference problem. We estimate the decay of the error, when the total variation of the control grows. Finally we prove a regularization result for state-constrained problems and for hybrid systems.

%\paragraph{Acknowledgments}
%Funding acknowledgments (if required)

\begin{thebibliography}{9} 
\bibitem{paper}
{\sc M.~Caponigro, R.~Ghezzi, B.~Piccoli, E.~Tr\'elat}, {\em Regularization of chattering phenomena via bounded variation control}, preprint 2013, arXiv:1303.5796.  

\bibitem{fuller}
{\sc A.T.~Fuller}, {\em Relay control systems optimized for various performance criteria}, Proceedings of the 1st {W}orld {C}ongress {IFAC}, {M}oscow, 1960, {B}utterworths, London (1961), pp.~510--519.  
\bibitem{zeno-numpb}
{\sc K.H.~Johansson, M.~Egerstedt, J.~Lygeros, S.~Sastry}, {\em On the regularization of Zeno hybrid automata}, Systems \& Control Letters, 38 (3), (1999), pp.~141--150.  
\bibitem{robbins}
{\sc H.~Robbins}, {\em Junction phenomena for optimal control with state-variable
              inequality constraints of third order}, Journal of Optimization Theory and Applications, 31 (1), (1980), pp.~85--99.  


\end{thebibliography}

\end{document}

@unpublished{fuller,
AUTHOR={Caponigro, Marco and Ghezzi, Roberta and Piccoli, B. and Tr\'elat, E.},
TITLE={{R}egularization of chattering phenomena via bounded variation control},
NOTE={preprint 2013, arXiv:1303.5796},
}		
AUTHOR = {Robbins, H.},
     TITLE = {Junction phenomena for optimal control with state-variable
              inequality constraints of third order},
   JOURNAL = {J. Optim. Theory Appl.},
  FJOURNAL = {Journal of Optimization Theory and Applications},
    VOLUME = {31},
      YEAR = {1980},
    NUMBER = {1},
     PAGES = {85--99},
    
    
    @incollection {fuller,
    AUTHOR = {Fuller, A. T.},
     TITLE = {Relay control systems optimized for various performance criteria},
BOOKTITLE = {Proceedings of the 1st {W}orld {C}ongress {IFAC}, {M}oscow, 1960, {B}utterworths, London (1961)
},
PAGES={510--519},
}

  AUTHOR = {Johansson, Karl Henrik and Egerstedt, Magnus and Lygeros, John
              and Sastry, Shankar},
     TITLE = {On the regularization of {Z}eno hybrid automata},
      NOTE = {Hybrid control systems},
   JOURNAL = {Systems Control Lett.},
  FJOURNAL = {Systems \& Control Letters},
    VOLUME = {38},
      YEAR = {1999},
    NUMBER = {3},
     PAGES = {141--150},

