We consider a nonlinear control system depending on two controls \begin{document} $u$ \end{document} and \begin{document} $v$ \end{document} , with dynamics affine in the (unbounded) derivative of \begin{document} $u$ \end{document} , and \begin{document} $v$ \end{document} appearing initially only in the drift term. Recently, motivated by applications to optimization problems lacking coercivity, Aronna and Rampazzo [ 1 ] proposed a notion of generalized solution \begin{document} $x$ \end{document} for this system, called limit solution, associated to measurable \begin{document} $u$ \end{document} and \begin{document} $v$ \end{document} , and with $u$ of possibly unbounded variation in \begin{document} $[0, T]$ \end{document} . As shown in [ 1 ], when \begin{document} $u$ \end{document} and \begin{document} $x$ \end{document} 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. Rishel [ 25 ], Warga [ 27 ] and Bressan and Rampazzo [ 8 ]). In [ 24 ] we extended this correspondence to BV \begin{document}$_{loc}$ \end{document} inputs \begin{document} $u$ \end{document} and trajectories (with bounded variation just on any \begin{document} $[0, t]$ \end{document} with \begin{document} $t ). Here, 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 \begin{document} $x$ \end{document} , for which such a minimum exists. As a first result, we prove that extended BV (respectively, BV \begin{document}$_{loc}$ \end{document} ) simple limit solutions and BV (respectively, BV \begin{document}$_{loc}$ \end{document} ) simple limit solutions coincide. Then we consider dynamics where the ordinary control \begin{document} $v$ \end{document} 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.