Entropy dissipation rates in thermodynamically realizable systems

In biological systems, a key concept of interest is the energy required to induce changes in the state of a system. The modern approach to this problem is to identify the energy cost of such control with entropy dissipation. For systems with deterministic control, the lower bounds on entropy dissipation are well known. However such analyses tend to ignore the cost associated with the method of control which becomes crucial at the biological scale. Here we extend such analyses to take into account the entropy cost of exerting control on a system when the control itself is stochastic in nature and subject to fluctuations. Using the formalism of stochastic thermodynamics, we find an intriguing non-trivial unification of previous results for a realizably controlled non-equilibrium steady state system driven at a finite rate around a loop in thermodynamic space. In particular we show that for a model system, the lower dissipation bound is almost the sum of two previously found bounds. Despite this, our result suggests that even for an infinitely long control protocol, it is still impossible to reach the adiabatic limit for stochastic systems.