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\title[Limit solutions]{"Limit solutions" for control systems}

\begin{document}

\author[M.S. Aronna]{M.- Soledad Aronna}
\address{M.S. Aronna, Rosario, Argentina}
\email{aronna@cmap.polytechnique.fr}

\author[F. Rampazzo]{Franco Rampazzo}
\address{F. Rampazzo, Dipartimento di Matematica ,
Universit\`a di Padova\\ Padova  35121, Italy}
\email{rampazzo@math.unipd.it}








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\begin{abstract}
{For a control Cauchy problem
$$\dot x= {f}(t,x,u,v) +\sum_{\alpha=1}^m g_\alpha(x) \dot u_\alpha\quad x(a)=\bar x,  $$
on an interval $[a,b]$,
 we propose  a notion of  {\em limit solution} $x$ that verifies  the following properties: i) $x$ is defined  for $\L^1$ {(impulsive)}   inputs  $u$  and for standard, bounded measurable, controls  $v$; ii) in the commutative case (i.e. when $[g_{\alpha},g_{\beta}]\equiv 0,$ for all $\alpha,\beta=1,\dots,m$), $x$ coincides with the  solution constructed via multiple fields' rectification;
iii) $x$ subsumes  former concepts of solution valid for the generic, noncommutative case.
 In particular,  when $u$ has  bounded variation, we investigate the relation between  limit solutions and (single-valued)  graph completion solutions.
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Furthermore, we prove consistency with the classical Carath\'eodory solution when $u$ and $x$ are absolutely continuous.  Some generalizations are under investigation, e.g. to the case where the dynamics is polynomial in $\dot u$.}
\end{abstract}
\end{document}