  The fluctuation theorems have remained one of the cornerstones in the study
of systems that are driven far out of equilibrium, and they provide strong
constraints on the fraction of trajectories that behave atypically in light of
the second law. They have mainly been derived for a predetermined external
drive applied to the system. However, to improve the efficiency of a process,
one needs to incorporate protocols that are modified by receiving feedbacks
about the recent state of the system, during its evolution. In such a case, the
forms of the conventional fluctuation theorems get modified, the correction
term involving terms that depend on the way the reverse/conjugate process is
defined, namely, the rules of using feedback in order to generate the exact
time-reversed/conjugate protocols. We show in this paper that this can be done
in a large number of ways, and in each case we would get a different expression
for the correction terms. This would in turn lead to several lower bounds on
the mean work that must be performed on the system, or on the entropy changes.
Here we analyze a form of the extended fluctuation theorems that involves the
efficacy parameter, and find that this form gives rise to a lower bound for the
mean work that retains a consistent physical meaning regardless of the design
of feedback along the conjugate process, as opposed to the case of the
previously mentioned form of the modified fluctuation theorems.
