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We present a novel reformulation of nonsmooth differential equations withstate jumps which enables their easier simulation and use in optimal controlproblems without the need of using integer variables. The main idea is tointroduce an auxiliary differential equation to mimic the state jump map.Thereby, also a clock state is introduced which does not evolve during theruntime of the auxiliary system. The pieces of the trajectory that correspondto the parts when the clock state was evolving recover the solution of theoriginal system with jumps. Our reformulation results in nonsmooth ordinarydifferential equations where the discontinuity is in the first time derivativeof the trajectory, rather than in the trajectory itself. This class of systemsis easier to handle both theoretically and numerically. We provide numericalexamples demonstrating the ease of use of this reformulation in both simulationand optimal control. In the optimal control example a single call of anonlinear programming (NLP) solver yields the same solution as a multi-stageformulation, without the need for exploring the optimal number of stages byenumeration or heuristics.

A Time-Freezing Approach for Numerical Optimal Control of Nonsmooth Differential Equations with State Jumps