  This is the first part of four series papers, aiming at the problem of
actuator dynamics compensation for linear systems. We consider the
stabilization of a type of cascade abstract linear systems which model the
actuator dynamics compensation for linear systems where both the control plant
and its actuator dynamics can be infinite-dimensional. We develop a systematic
way to stabilize the cascade systems by a full state feedback. Both the
well-posedness and the exponential stability of the resulting closed-loop
system are established in the abstract framework. A sufficient condition of the
existence of compensator for ordinary differential equation (ODE) with partial
differential equation (PDE) actuator dynamics is obtained. The feedback design
is based on a novelly constructed upper-block-triangle transform and the
Lyapunov function design is not needed in the stability analysis. As
applications, an ODE with input delay and an unstable heat equation with ODE
actuator dynamics are investigated to validate the theoretical results. The
numerical simulations for the unstable heat system are carried out to validate
the proposed approach visually.
