Optimal Control of Uniformly Heated Granular Fluids in Linear Response

We present a detailed analytical investigation of the optimal control of uniformly heated granular gases in the linear regime. The intensity of the stochastic driving is therefore assumed to be bounded between two values that are close, which limits the possible values of the granular temperature to a correspondingly small interval. Specifically, we are interested in minimising the connection time between the non-equilibrium steady states (NESSs) for two different values of the granular temperature by controlling the time dependence of the driving intensity. The closeness of the initial and target NESSs make it possible to linearise the evolution equations and rigorously—from a mathematical point of view—prove that the optimal controls are of bang-bang type, with only one switching in the first Sonine approximation. We also look into the dependence of the optimal connection time on the bounds of the driving intensity. Moreover, the limits of validity of the linear regime are investigated.