  We investigate the influence of environment noise on the control cost and the
quantum speed limit time (QSLT) in the process of almost exact state
transmission (AEST) through a spin chain under pulse control. The chain is
immersed in its surrounding non-Markovian, finite temperature heat baths. We
find that AEST can be realized in weak system-bath coupling, low temperature,
and strong non-Markovian baths under effective external control.
Correspondingly, the control cost and QSLT increases with increasing bath
temperature and coupling strength. It is noticeable that non-Markovianity from
the baths can be helpful to reduce the control cost and shorten the QSLT.
Furthermore, we find that there exists a trade-off between the control cost and
transmission fidelity and higher fidelity requires higher cost. In addition,
the minimum control cost has been found to obtain certain transmission
fidelity.
