  We present a novel reformulation of nonsmooth differential equations with
state jumps which enables their easier simulation and use in optimal control
problems without the need of using integer variables. The main idea is to
introduce an auxiliary differential equation to mimic the state jump map.
Thereby, also a clock state is introduced which does not evolve during the
runtime of the auxiliary system. The pieces of the trajectory that correspond
to the parts when the clock state was evolving recover the solution of the
original system with jumps. Our reformulation results in nonsmooth ordinary
differential equations where the discontinuity is in the first time derivative
of the trajectory, rather than in the trajectory itself. This class of systems
is easier to handle both theoretically and numerically. We provide numerical
examples demonstrating the ease of use of this reformulation in both simulation
and optimal control. In the optimal control example a single call of a
nonlinear programming (NLP) solver yields the same solution as a multi-stage
formulation, without the need for exploring the optimal number of stages by
enumeration or heuristics.
