  We look into the minimisation of the connection time between non-equilibrium
steady states. As a prototypical example of an intrinsically non-equilibrium
system, a driven granular gas is considered. For time-independent driving, its
natural time scale for relaxation is characterised from an empirical -- the
relaxation function -- and a theoretical -- the recently derived classical
speed limits -- point of view. Using control theory, we find that bang-bang
protocols -- comprising two steps, heating with the largest possible value of
the driving and cooling with zero driving -- minimise the connecting time. The
bang-bang time is shorter than both the empirical relaxation time and the
classical speed limit: in this sense, the natural time scale for relaxation is
beaten. Information theory quantities stemming from the Fisher information are
also analysed over these optimal protocols. The implementation of the bang-bang
processes in numerical simulations of the dynamics of the granular gas show an
excellent agreement with the theoretical predictions. Moreover, general
implications of our results are discussed for a wide class of driven
non-equilibrium systems. Specifically, we show that analogous bang-bang
protocols, with a number of bangs equal to the number of relevant physical
variables, give the minimum connecting time under quite general conditions.
