---
abstract: 'We establish a general theory of feedback control on classical stochastic thermodynamic systems, and generalize nonequilibrium equalities such as the fluctuation theorem and the Jarzynski equality in the presence of feedback control with multiple measurements. Our results are generalizations of the previous relevant works to the situations with general measurements and multi-heat baths. The obtained equalities involve additional terms that characterize the information obtained by measurements or the efficacy of feedback control. A generalized Szilard engine and a feedback-controlled ratchet are shown to satisfy the derived equalities.'
author:
- 'Takahiro Sagawa$^{1,2}$'
- 'Masahito Ueda$^{3,4}$'
title: Nonequilibrium Thermodynamics of Feedback Control
---

Introduction
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Since the mid-twentieth century, feedback control has played crucial roles in science and engineering [@Doyle; @Astrom]. Here, “feedback” means that a control protocol depends on measurement outcomes obtained from the controlled system. Recently, feedback control has become increasingly important in terms of nonequilibrium physics, due to at least the following two reasons.

First of all, stochastic aspects of thermodynamics [@Sekimoto; @Bustamante; @Seifert1] have become important due to recent theoretical and experimental developments. Theoretically, a number of nonequilibrium equalities such as the fluctuation theorem and the Jarzynski equality [@Evans; @Gallavotti; @Jarzynski1; @Crooks1; @Crooks2; @Spohn; @Maes1; @Maes2; 