Based on the LMI-technique, and special modification of the Petersen lemma, we propose a regular approach to the robust stabilization of bilinear control systems via a linear static state feedback. In the state space of the system, we design a so-called robust stabilizability ellipsoid, such that the trajectory of the closed-loop system, starting inside the ellipsoid asymptotically tends to zero for all admissible system uncertainties. The natural development of this approach allows to effectively design an approximation of the robust stabilizability domain of bilinear control systems. The proposed approach is based on the solution of convex optimization problems, yet allows for the construction of nonconvex approximations of robust stabilizability domains of uncertain bilinear systems. The approach is easily implementable from the computational point of view. It is based on the solution of convex optimization problems; however, it leads to nonconvex approximations of stabilizability domains of bilinear systems. Moreover, the proposed approach can be extended to the discrete-time case, and to other robust statements of the problem.