  A specific class of stochastic heat engines driven cyclically by
time-dependent potential, which is defined in the half-line ($0<x<+\infty$), is
analysed. For such engines, most of their physical quantities can be obtained
explicitly, including the entropy and internal energy of the heat engine, as
well as output work (power) and heat exchange with the environment during a
finite time interval. The optimisation method based on the external potential
to reduce {\it irreversible} work and increase energy efficiency is presented.
With this optimised potential, efficiency $\eta^*$ and its particular value at
maximum power $\eta^*_{\rm EMP}$ are calculated and discussed briefly.
