We consider random switching between finitely many vector fields leaving positively invariant a compact set. Recently, Li, Liu and Cui showed in  that if one the vector fields has a globally asymptotically stable (G.A.S.) equilibrium from which one can reach a point satisfying a weak Hörmander-bracket condition, then the process converges in total variation to a unique invariant probability measure. In this note, adapting the proof in  and using results of , the assumption of a G.A.S. equilibrium is weakened to the existence of an accessible point at which a barycentric combination of the vector fields vanishes. Some examples are given which demonstrate the usefulness of this condition.
"A user-friendly condition for exponential ergodicity in randomly switched environments." Electron. Commun. Probab. 23 1 - 12, 2018. https://doi.org/10.1214/18-ECP148