State feedback control of bifurcations with quadratic or cubic degeneracy is addressed. Based on normal forms and invariants of nonlinear control systems, the classification of bifurcations for systems with a single uncontrollable mode is obtained. Using invariants, characterizations of stability are derived for a family of bifurcations, including saddle-node bifurcation and transcritical bifurcation in this paper. Bifurcations in systems under feedbacks which are not zero at the critical point are also addressed. In the case of a saddle-node bifurcation, continuous but not differential feedbacks are introduced to remove the bifurcation and to achieve the stability.