On 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 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 by Aronna and Rampazzo, 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, Warga and Bressan and Rampazzo). In a previous paper we extended this correspondence to BVloc 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, BVloc) simple limit solutions and BV (respectively, BVloc) 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