  Milky Way dwarf satellites are unique objects that encode the early structure
formation and therefore represent a window into the high redshift Universe. So
far, their study was conducted using electromagnetic waves only. The future
Laser Interferometer Space Antenna (LISA) has the potential to reveal Milky Way
satellites in gravitational waves emitted by double white dwarf (DWD) binaries.
We investigate gravitational wave (GW) signals detectable by LISA as a possible
tool for the identification and characterisation of the Milky Way satellites.
We use the binary population synthesis technique to model the population of
DWDs in dwarf satellites and we assess the impact on the number of LISA
detections when making changes to the total stellar mass, distance, star
formation history and metallicity of satellites. We calibrate predictions for
the known Milky Way satellites on their observed properties. We find that DWDs
emitting at frequencies $\gtrsim 3\,$mHz can be detected in Milky Way
satellites at large galactocentric distances. The number of these high
frequency DWDs per satellite primarily depends on its mass, distance, age and
star formation history, and only mildly depends on the other assumptions
regarding their evolution such as metallicity. We find that dwarf galaxies with
$M_\star>10^6\,$M$_{\odot}$ can host detectable LISA sources with a number of
detections that scales linearly with the satellite's mass. We forecast that out
of the known satellites, Sagittarius, Fornax, Sculptor and the Magellanic
Clouds can be detected with LISA. As an all-sky survey that does not suffer
from contamination and dust extinction, LISA will provide observations of the
Milky Way and dwarf satellites galaxies valuable for Galactic archaeology and
near-field cosmology.
