Dynamical systems with discontinuous solutions and problems with unbounded derivatives

V. I. Gurman suggested a description of discontinuous solutions in terms of systems with unbounded derivatives. The idea was in the usage of an auxiliary system of ordinary differential equations including the recession cone of the velocities set. It was useful for inclusion discontinuous functions into the set of admissible solutions, however, it became clear later that such a description is not only correct, but it gives also the unique in some sense representation of solutions which guaranties the existence of a solution for corresponding variational problems.
In this article, we describe the subsequent development of this methodology for variational problems where the solutions discontinuities appear naturally as a result of the impacts against the rigid surfaces. We give an illustration of the singular spatio-temporal transformation technique for problems of impact with friction. As an example, we consider a system with the Painlevé paradox, namely, a mathematical formalization of oblique impact, where the contact law is described by a viscous-elastic Kelvin–Voigt model, and the contact termination is defined as a moment when the supporting force vanishes.