  Resetting, in which a system is regularly returned to a given state after a
fixed or random duration, has become a useful strategy to optimize the search
performance of a system. While earlier theoretical frameworks focused on
instantaneous resetting, wherein the system is directly teleported to a given
state, there is a growing interest in physical resetting mechanisms that
involve a finite return time. However employing such a mechanism involves cost
and the effect of this cost on the search time remains largely unexplored. Yet
answering this is important in order to design cost-efficient resetting
strategies. Motivated from this, we present a thermodynamic analysis of a
diffusing particle whose position is intermittently reset to a specific site by
employing a stochastic return protocol with external confining trap. We show
for a family of potentials $U_R(x) \sim |x|^{m}$ with $m>0$, it is possible to
find optimal potential shape that minimises the expected first-passage time for
a given value of the thermodynamic cost, i.e. mean work. By varying this value,
we then obtain the Pareto optimal front, and demonstrate a trade-off relation
between the first-passage time and the work done.
