of thermodynamics. Feedback control may be regarded as a powerful tool to control thermodynamic systems. Since the crucial quantity is the information that is obtained to be used for feedback control, we may regard the Szilard-type heat engine as “information heat engine.” Recently, such an information heat engine was realized experimentally by using a colloidal particle [@Toyabe3].

In this paper, we formulate a general theory of feedback control on stochastic thermodynamic systems. In particular, we extend recent theoretical results on the generalizations of the fluctuation theorem and the Jarzynski equality [@Sagawa-Ueda3] to the situations in which the measurement and feedback control are non-Markovian and there are multi-heat baths. Our results serve as the fundamental building blocks of information heat engines.

This paper is organized as follows.

In Sec. II, we briefly review the framework of stochastic thermodynamics in a general setup. We discuss classical stochastic systems that are in general non-Markovian and in contact with multi-heat baths. We discuss the concept of entropy production and the detailed fluctuation theorem as our starting point. Because they are general properties of nonequilibrium systems, our formulations and results in the following sections are not restricted to Langevin systems but applicable to any classical 