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

## TL;DR

This paper studies generalized solutions for nonlinear control systems with impulsive controls, introducing an extended notion of limit solutions to ensure the existence of optimal solutions, and establishing their equivalence with graph completion solutions in certain cases.

## Contribution

It introduces an extended limit solution concept for impulsive control systems, ensuring existence of minima, and proves their equivalence with graph completion solutions under specific conditions.

## Key findings

- Extended limit solutions coincide with original limit solutions for BV controls.
- In systems with controls appearing in non-drift terms, extended solutions match graph completion solutions.
- The new framework guarantees the existence of optimal solutions in impulsive control problems.

## Abstract

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.