The Gaia ESO Public Spectroscopic Survey (GES) is providing the astronomical community with high-precision measurements of many stellar parameters including radial velocities (RVs) of stars belonging to several young clusters and star-forming regions. One of the main goals of the young cluster observations is to study of their dynamical evolution and provide insight into their future, revealing if they will eventually disperse to populate the field, rather than evolve into bound open clusters. In this paper we report the analysis of the dynamical state of L1688 in the $rho$~Ophiuchi molecular cloud using the dataset provided by the GES consortium. We performed the membership selection of the more than 300 objects observed. Using the presence of the lithium absorption and the location in the Hertzspung-Russell diagram, we identify 45 already known members and two new association members. We provide accurate RVs for all 47 confirmed members.A dynamical analysis, after accounting for unresolved binaries and errors, shows that the stellar surface population of L1688 has a velocity dispersion $sigma sim$1.14$pm$0.35 km s$^{-1}$ that is consistent with being in virial equilibrium and is bound with a $sim$80% probability. We also find a velocity gradient in the stellar surface population of $sim$1.0 km s$^{-1}$pc$^{-1}$ in the northwest/southeast direction, which is consistent with that found for the pre-stellar dense cores, and we discuss the possibility of sequential and triggered star formation in L1688.
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