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We consider a control system with dynamics which are affine in the(unbounded) derivative of the control $u$. We introduce a notion of generalizedsolution $x$ on $[0,T]$ for controls $u$ of bounded total variation on $[0,t]$for every $t<T$, but of possibly infinite variation on $[0,T]$. This solutionhas a simple representation formula based on the so-called graph completionapproach, originally developed for BV controls.  We prove the well-posedness of this generalized solution by showing that $x$is a limit solution, that is the pointwise limit of regular trajectories of thesystem. In particular, we single out the subset of limit solutions which is inone-to-one correspondence with the set of generalized solutions. The controlsthat we consider provide the natural setting for treating some questions on thecontrollability of the system and some optimal control problems with endpointconstraints and lack of coercivity.

Unbounded variation and solutions of impulsive control systems