On ${\mathcal L}^1$ limit solutions in impulsive control

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, Aronna and Rampazzo [1] proposed a notion of generalized solution $x$ for this system, called 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. Rishel [25], Warga [27] and Bressan and Rampazzo [8]). In [24] we extended this correspondence to BV$_{loc}$ inputs $u$ and trajectories (with bounded variation just on any $[0, t]$ with $t<T$). 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 $x$, for which such a minimum exists. As a first result, we prove that extended BV (respectively, BV$_{loc}$) simple limit solutions and BV (respectively, BV$_{loc}$) simple limit solutions coincide. 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.