We prove the existence and uniqueness of solutions in Sobolev spaces for the Moore-Greitzer nonlinear partial differential equation (PDE) model for compression system instabilities with mild conditions on the shape of the compressor characteristic and on the throttle control. To achieve this, the model is reformulated as an evolution equation on a Banach space. Using this new representation, we design a backstepping control of the model. Global stabilization of any axisymmetric equilibrium to the right of the peak of the compressor characteristic is achieved. We also prove that the dynamics can be restricted to the small neighborhood of the point on the left of the peak of the compressor characteristic. Thus, it is possible to restrict the magnitude of stall to arbitrary small values. In addition, finite-dimensional Galerkin projections of the partial differential equation model are studied. It is shown that truncated control laws stabilize truncated models. Numerical simulations of the model with and without control are presented.