On the stabilizability and controllability of a degenerate controlled system in Banach spaces

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.