x_n )\Delta t,$$ $$A_{n+1} - A_n = \left(  \frac{-2K A_n + 2k_{\rm B}T} {\eta}- \frac{A_n}{R} \right)\Delta t, 
\label{R}$$ where $A_n$ is a time-dependent real number and Eq. (\[R\]) is a discretized version of the Riccati equation. By using the Kalman estimator $\hat x_n$, the optimal control protocol [@Bertsekas] is given by $$\lambda_n = - C_n \hat x_n,$$ where $C_n$ is a pre-determined constant depending on the target of the optimal control. We note that the optimal control is a non-Markovian control as $\lambda_n = \lambda_n (Y_n)$, because we use all of $Y_n = (y_0, y_1, \cdots, y_n)$ to calculate $\hat x_n$. The generalized Jarzynski equality for this situation has been discussed in Ref. [@Suzuki2].

Probability Distributions with Feedback
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We discuss the probability distributions with feedback control in general. Under the condition that we fix control protocol $\Lambda_N (Y_N)$ with $Y_N$ being fixed, the conditional probability of realizing $X_N$ is given by $$P[X_n | \Lambda_n (Y_{n-1})] = P_0 [X_0] \prod_{k=0}^{k-1} P[x_{k+1} | X_k, \lambda_k (Y_k)],
\label{prob_tr_f}$$ which corresponds to Eq. (\[prob\_tr1\]). We note that, in the expression in Eq. (\[prob\_tr\_f\]), we do not omit the notation $\Lambda_N (Y_{N-1})$ because its $Y_{N-1}$-dependence is crucial. We also write $$P[X_n | x_0, \Lambda_n 