The article presents an approach to simultaneous stabilization of a finite family of linear dynamical plants. Two main issues are considered: finding the existence conditions for a single controller and an algorithm to construct such a controller if it exists. The approach proposed in this article is based on the properties of affine transformations of the controller parameter space into the coefficient space of characteristic polynomials of the stabilized plants. A general scheme is presented for analyzing simultaneous stabilizability of finite families of plants and a numerical algorithm is investigated to find a stabilizing controller by the interval analysis method.