$\langle \Delta R_n \Delta R_{n'} \rangle = R \delta_{nn'}\Delta t$ ($R>0$). The conditional probability of obtaining outcome $y_n$ is given by $$P[y_n | x_n] \propto \exp \left[  - \frac{\Delta t}{2R} (y_n -x_n)^2 \right].$$

The feedback protocol can be written as $\lambda_n (y_0, y_1, \cdots, y_n)$ in general. The work performed on the system is then given by $$\begin{split}
W_n &:= V(x_n, \lambda_{n+1}) - V (x_n, \lambda_n) \\
&= \frac{\partial V}{\partial \lambda} \Delta \lambda_n + o(\Delta t), 
\end{split}$$ where $$\Delta \lambda_n := \lambda_{n+1} (y_0, y_1, \cdots, y_{n+1}) - \lambda_n (y_0, y_1, \cdots, y_n).$$ In particular, if the feedback is Markovian, $\lambda_n$ is given by $\lambda_n (y_n)$. Then, $\Delta \lambda_n = \lambda_{n+1} (y_{n+1}) - \lambda_n (y_n)$ can be written as $$\Delta \lambda_n = \frac{\partial \lambda}{\partial t} \Delta t + \frac{\partial \lambda}{\partial y} \Delta y_n + \frac{1}{2}\frac{\partial^2 \lambda}{\partial y^2} \Delta y_n^2,
\label{lambda}$$ where $\Delta y_n := y_{n+1} - y_n$. The first term on the right-hand side of Eq. (\[lambda\]) arises from a change in $\lambda$ by the pre-fixed protocol, while the second and third terms are induced by the feedback control.

We next consider the Kalman filter and the optimal control. As a special case of Eq. (\[Langevin\]), we consider a discretized linear 