A combination of theoretical and computational nonlinear analysis techniques are used to study the scenario of bifurcations responsible for the initiation of rotating stall in an axial flow compressor model. It is found that viscosity tends to damp higher-frequency modes and so results in a sequence of bifurcations along the uniform-flow solution branch to stall cells of different mode number. Lower-mode stalled flow solutions are born in subcritical bifurcations, meaning that these equilibria will be unstable for small amplitudes. Secondary bifurcations, however, can render them stable, leading to hysteresis. Using throttle position as a control, we find that while the stall bifurcations are not linearly stabilizable, nonlinear state feedback of the first mode amplitude will reduce the hysteresis. This improves the nonlinear stability of the compression system near the stall margin.