A relaxation result for unbounded control system with hysteresis

We consider a control system of two ODEs describing a hysteresis effect. The term unbounded in the title refers to the fact that the multivalued mapping giving the control constraints may admit unbounded values. Being at the same time nonconvex valued, we convexify the constraint and we approximate solutions of thus obtained relaxed problem by those of the given system. The property of Lipschitz continuity of the multivalued mapping traditionally employed for such kind of relaxation problems is weakened to a more appropriate in the context of unbounded valued multifunctions “truncated” version of it.