We consider the design problem for bilinear control systems subjected to arbitrary bounded exogenous disturbances. A procedure for the construction of the stabilizability ellipsoid and domain of stabilizability for bilinear control systems is proposed and its efficiency is proved. The main tools are the linear matrix inequality technique and the apparatus of quadratic Lyapunov functions. This simple yet general approach is of great potential; for instance, it can be generalized to the discrete-time case, and to the various robust statements of the problem.