  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 limit solution $x,$ verifying the following properties: i)
$x$ is defined for $\mathcal{L}^1$ (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 $\alpha,\beta=1,...,m$), $x$
coincides with the solution one can obtain via the change of coordinates that
makes the $g_\alpha$ simultaneously constant; 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.
Furthermore, we prove consistency with the classical Carath\'eodory solution
when $u$ and $x$ are absolutely continuous.
  Even though some specific problems are better addressed by means of special
representations of the solutions, we believe that various theoretical issues
call for a unified notion of trajectory. For instance, this is the case of
optimal control problems, possibly with state and endpoint constraints, for
which no extra assumptions (like e.g. coercivity, bounded variation,
commutativity) are made in advance.
