  In this paper we constrain four alternative models to the late cosmic
acceleration in the Universe: Chevallier-Polarski-Linder (CPL), interacting
dark energy (IDE), Ricci holographic dark energy (HDE), and modified polytropic
Cardassian (MPC). Strong lensing (SL) images of background galaxies produced by
the galaxy cluster Abell $1689$ are used to test these models. To perform this
analysis we modify the LENSTOOL lens modeling code. The value added by this
probe is compared with other complementary probes: Type Ia supernovae (SNIa),
baryon acoustic oscillations (BAO), and cosmic microwave background (CMB). We
found that the CPL constraints obtained of the SL data are consistent with
those estimated using the other probes. The IDE constraints are consistent with
the complementary bounds only if large errors in the SL measurements are
considered. The Ricci HDE and MPC constraints are weak but they are similar to
the BAO, SNIa and CMB estimations. We also compute the figure-of-merit as a
tool to quantify the goodness of fit of the data. Our results suggest that the
SL method provides statistically significant constraints on the CPL parameters
but weak for those of the other models. Finally, we show that the use of the SL
measurements in galaxy clusters is a promising and powerful technique to
constrain cosmological models. The advantage of this method is that
cosmological parameters are estimated by modelling the SL features for each
underlying cosmology. These estimations could be further improved by SL
constraints coming from other galaxy clusters.
