/ P[Y_n])$. In the presence of feedback control, however, this is not true (i.e., $I_{\rm c} \neq I$), as we will see later.

Feedback Control
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In this section, we formulate feedback control on nonequilibrium dynamics.

Formulation
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Feedback control implies that protocol $\Lambda_N$ depends on measurement outcomes $Y_N$ (see FIG. 2). On the other hand, without feedback control, control protocols are predetermined and independent of the measurement outcomes, as is the case for the setup of the original fluctuation theorem and Jarzynski equality.

![Feedback control on nonequilibrium dynamics. The control parameter is denoted as $\lambda$, the point of the phase space of the system as $x$, and the outcome of measurement on the system as $y$. Parameter $\lambda$ depends on $y$ through the real-time feedback control. ](Fig2.eps){width="75mm"}

When the system is subject to feedback control, $\lambda_n$ can depend on measurement outcomes that are obtained until $t_n$, while $\lambda_n$ cannot depend on any measurement outcome that is obtained after time $t_n$ due to causality. We introduce notation $\lambda_n (Y_n)$, which means that the value of $\lambda$ at time $t_n$ is determined by $Y_n$. We write $\Lambda_n (Y_{n-1}) := (\lambda_0 (Y_0), \lambda_1 (Y_1), \cdots, \lambda_{n-1} (Y_{n-1}))$.

If $\lambda_n$ only depends on 