Self-sustained thermoacoustic oscillations resulting from a dynamic coupling between unsteady heat release and flow perturbations are detrimental to engine systems such as boilers, land-based gas turbines and furnaces. In this work, mitigating premixed flame-sustained thermoacoustic oscillations in a combustor is studied by actively passive control of a Helmholtz resonator with an oscillating volume. For this, a numerical model of a thermoacoustic combustor with a Helmholtz resonator attached is developed. As a feedback control technique is implemented on the combustion system to optimize the resonator's damping effect, combustion-driven oscillations are successfully mitigated by approximately 30 dB. In addition, the dynamic response of the flame to oncoming acoustic disturbances is studied. This is achieved by numerically solving a nonlinear G-equation tracking flame front, as the disturbances with multiple frequencies are imposed. Flame transfer function is then derived via linearizing the flame model to obtain analytical results. Comparison is then made between the analytical and numerical results. It is shown from the transfer function analysis that the unsteady heat release linearly depends on the oncoming flow disturbance. However, the numerical results reveal that the flame response is nonlinear. Furthermore, the flame responds strongly to acoustic disturbances at a lower frequency and it behaves like a 'low-pass' filter. The flame speed is also found to play an important role in determining the unsteady heat release. Finally, to validate the effectiveness and performance of the feedback control, a Helmholtz resonator with a controllable oscillating diaphragm is implemented on a Y-shaped Rijke-type thermoacoustic system. Sound pressure level is shown to be reduced by more than 60 dB via tuning the gain and phase of the vibration of the loudspeaker diaphragm. The present investigation opens up an alternative applicable means to stabilize engine systems via minimizing thermoacoustic oscillations.