Search for stochastic gravitational-wave background from string cosmology with Advanced LIGO and Virgo's O1$\sim$O3 data

String cosmology models predict a relic background of gravitational-wave (GW) radiation in the early universe. The GW energy spectrum of radiated power increases rapidly with the frequency, and therefore it becomes a potential and meaningful observation object for high-frequency GW detector. We focus on the stochastic background generated by superinflation in string theory and search for such signal in the observing data of Advanced LIGO and Virgo O1$\sim$O3 runs in a Bayesian framework. We do not find the existence of the signal, and thus put constraints on the GW energy density. Our results indicate that at $f=100\text{Hz}$, the fractional energy density of GW background is less than $1.7\times {10}^{-8}$ and $2.1\times {10}^{-8}$ for dilaton-string and dilaton only cases respectively, and further rule out the parameter space restricted by the model itself due to the non-decreasing dilaton and stable cosmology background ($\beta$ bound).