Absorbing airborne noise at frequencies below 300 Hz is a particularly vexing problem due to the absence of natural sound absorbing materials at these frequencies. The prevailing solution for low-frequency sound absorption is the use of passive narrow-band resonators, whose absorption level and bandwidth can be further enhanced using nonlinear effects. However, these effects are typically triggered at high intensity levels, without much control over the form of the nonlinear absorption mechanism. In this study, we propose, implement, and experimentally demonstrate a nonlinear active control framework on an existing experimental (linear) electroacoustic resonator prototype, allowing for unprecedented control over the form of non-linearity, and arbitrarily low absorption intensity thresholds. More specifically, the proposed architecture combines a linear feedforward control on the front pressure through a first microphone located at the front face of the loudspeaker, and a nonlinear feedback on the membrane displacement estimated through the measurement of the pressure inside the back cavity with a second microphone located in the enclosure. It is experimentally shown that even at a weak excitation level, it is possible to observe and control the nonlinear behaviour of the system. Taking the cubic nonlinearity as an example, we demonstrate numerically and experimentally that in the low frequency range ($[\SI{50}{Hz}, \SI{600}{Hz}]$), the nonlinear control law allows improving the sound absorption performance, i.e. enlarging the bandwidth of optimal sound absorption while increasing the maximal absorption coefficient value. The reported experimental methodology can be extended to implement various types of hybrid linear and/or nonlinear controls, thus opening new avenues for managing wave nonlinearity and achieving non-trivial wave phenomena.