  We consider a nonlinear control system depending on two controls u and v,
with dynamics affine in the (unbounded) derivative of u, and v appearing
initially only in the drift term. Recently, motivated by applications to
optimization problems lacking coercivity, [1] proposed a notion of generalized
solution x for this system, called {\it limit solution,} associated to
measurable u and v, and with u of possibly unbounded variation in [0,T]. As
shown in [1], when u and x have bounded variation, such a solution (called in
this case BV simple limit solution) coincides with the most used graph
completion solution (see e.g. [6]). This correspondence has been extended in
[24] to BV_loc u and trajectories (with bounded variation just on any [0,t]
with t<T). Starting with an example of optimal control where the minimum does
not exist in the class of limit solutions, we propose a notion of extended
limit solution x, for which such a minimum exists. As a first result, we prove
that extended and original limit solutions coincide in the special cases of BV
and BV_loc inputs u (and solutions). Then we consider dynamics where the
ordinary control v also appears in the non-drift terms. For the associated
system we prove that, in the BV case, extended limit solutions coincide with
graph completion solutions.
