The aim of this research is to generalize the famous Lyapunov theorem concerning the classical explicit differential systems (Continuous or Discrete) described by two abstract forms: x’(t) = Px(t) or xn+1 = Pxn, where P is an operator or a matrix if the space has finite dimension, in order to study the spectrum of the degenerate differential systems: Ax(t) = Bx(t), for all t≥0. Here A and B are two bounded operators acting in Banach spaces also the operator A is not invertible. Using some properties of the spectral theory for the operator pencil of the corresponding systems which is obtained by substituting x(t) = e^(λt).v into the homogenous above equation, and an appropriate conformal mapping we obtain important results can be applied to study the stabilizability and controllability of certain degenerate controlled systems.