Control of active field theories at minimal dissipation

Experimental progress enables the precise manipulation of active systems that dissipate energy to sustain collective phenomena without any equilibrium equivalent. An outstanding challenge is to rationalize how material properties can be optimally controlled by applying external perturbations. We build an optimization procedure for generic active field theories within a thermodynamically consistent formulation. Central to our approach is the distinction between the \emph{protocol heat}, which is dissipated only during manipulation, and the \emph{total heat}, which also accounts for the post-manipulation dissipation. We demonstrate that the latter generically features a global minimum with respect to the protocol duration. We deploy our versatile approach to an active theory of phase separation, and examine the scalings of the optimal protocol duration with respect to activity and system size. Remarkably, we reveal that the landscape of steady-state dissipation regulates the crossover between optimal control strategies for a finite duration.