  We systematically investigate the effect of strain and doping on the polar
metal phase in lithium osmate, LiOsO$_3$, using first-principles calculations.
We demonstrate that the polar metal phase in LiOsO$_3$ can be controlled by
biaxial strain. Based on density functional calculations, we show that a
compressive biaxial strain enhances the stability of the polar $R3c$ phase. On
the other hand, a tensile biaxial strain favors the centrosymmetric
$R\overline{3}c$ structure. Thus, strain emerges as a promising control
parameter over polar metallicity in this material. We uncover a strain-driven
quantum phase transition under tensile strain, and highlight intriguing
properties that could emerge in the vicinity of this polar to non-polar metal
transition. We examine the effect of charge doping on the polar metal phase. By
means of electrostatic doping as well as supercell calculations, we find that
screening from additional charge carriers, expected to suppress the polar
distortions, have only a small effect on them. Rather remarkably, and in
contrast to conventional ferroelectrics, the polar metal phase in LiOsO$_3$
remains robust against charge doping up to large doping values.
