  We consider a control system with dynamics which are affine in the
(unbounded) derivative of the control $u$. We introduce a notion of generalized
solution $x$ on $[0,T]$ for controls $u$ of bounded total variation on $[0,t]$
for every $t<T$, but of possibly infinite variation on $[0,T]$. This solution
has a simple representation formula based on the so-called graph completion
approach, originally developed for BV controls.
  We prove the well-posedness of this generalized solution by showing that $x$
is a limit solution, that is the pointwise limit of regular trajectories of the
system. In particular, we single out the subset of limit solutions which is in
one-to-one correspondence with the set of generalized solutions. The controls
that we consider provide the natural setting for treating some questions on the
controllability of the system and some optimal control problems with endpoint
constraints and lack of coercivity.
