{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, [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.