stochastic systems that satisfy the detailed fluctuation theorem.

In Sec. III, we formulate measurements on thermodynamic systems. We discuss multi-measurements including continuous measurements, and investigate the properties of the mutual information obtained by the measurements. In particular, we introduce the two kinds of mutual information $I$ and $I_{\rm c}$, which will be shown to play key roles in the discussion of feedback control.

In Sec. IV, we discuss feedback control on Markov and non-Markov processes, and investigate feedback control in terms of probability theory, where the causality of the measurement and feedback play a crucial role.

In Sec. V, we derive the main results of this paper. We generalize the nonequilibrium equalities to situations in which the system is subject to feedback control. In particular, we derive two types of generalizations of the fluctuation theorem and the Jarzynski equality. One involves a term concerning the mutual information, and the other involves a term of feedback efficacy. As corollaries, we derive the generalizations of the second law of thermodynamics and a fluctuation-dissipation relation.

In Sec. VI, we illustrate our general results by two examples: a generalized Szilard engine with measurement errors and a feedback-controlled ratchet [@Cao1; @Lopez; @Cao3]. We discuss the 