Nonequilibrium fluctuation theorem for systems under discrete and continuous feedback control

In the time reverse process of a feedback manipulated stochastic system, we allow performing measurements without violating causality. As a result we come across an entropy production due to the measurement process. This entropy production, in addition to the usual system and medium entropy production, constitutes the total entropy production of the combined system of the reservoir, the system, and the feedback controller. We show that this total entropy production of full system satisfies an integrated fluctuation theorem as well as a detailed fluctuation theorem as expected. We illustrate and verify this idea through explicit calculation and direct simulation in two examples.