$y_n$ as $\lambda_n (y_n)$, the feedback protocol is called Markovian. We note that the Markovness of feedback is independent of that of the dynamics or measurements. The Markovian feedback control is realized when the delay time of feedback is sufficiently smaller than the smallest time scale $\Delta t$ of the dynamics.

Overdamped Langevin System
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As a simple illustrative example, we discuss an overdamped Langevin system, whose equation of motion is given by $$\eta \frac{dx (t)}{dt} = -\frac{\partial V(x, \lambda)}{\partial x} + f(\lambda) + \sqrt{2 \eta k_{\rm B}T} \xi (t),
\label{Langevin}$$ where $\eta$ is the friction constant, $V(x, \lambda)$ is an external potential, $f(\lambda)$ is an external nonconservative force, and $\xi (t)$ is the Gaussian white noise satisfying $\langle \xi (t) \xi (t') \rangle = \delta (t-t')$. The detailed fluctuation theorem (\[fluctuation1\]) is still satisfied in the presence of a nonconservative force that violates the detailed balance, because Eq. (\[fluctuation1\]) can be derived from the local transition rate of the stochastic dynamics and is independent whether there is a global potential or not. We assume that the measurement is time-continuous and Markovian: $$y_n = x_n + \frac{\Delta R_n}{\Delta t },
\label{measurement_Langevin}$$ where $\Delta R_n$ is a white Gaussian noise with 