The Minkowski Wrap: A Relativistic Speed Limiter

In special relativity, a particle can experience constant acceleration, but at the same time, its motion is constrained by the velocity limit imposed by the speed of light c. Inspired by this principle, we propose a method for enforcing velocity bounds in control systems by replacing c with the maximum attainable speed of the system. We refer to this as the ``Minkowski wrap," the operation of deforming the phase portrait of a system so as to enforce desired speed limits. We apply this idea to shape the input generated by a state-feedback stabilizing controller and time-optimal controller considering controlling a double integrator system. By applying Pontryagin's Maximum Principle, we show that the time-optimal control of a wrapped double-integrator system is bang-bang. The proposed method transforms the classical double-integrator dynamics into a ``wrapped" system that respects velocity bounds without the need for clipping, offering an explicit nonlinear feedback control strategy conducive to safety applications.