  For linear control systems with bounded control range, chain controllability
properties are analyzed. It is shown that there exists a unique chain control
set and that it equals the sum of the control set around the origin and the
center Lyapunov space of the homogeneous part. For the proof, the linear
control system is extended to a bilinear control system on an augmented state
space. This system induces a control system on projective space. For the
associated control flow attractor-repeller decompositions are used to show that
the control system on projective space has a unique chain control set that is
not contained in the equator. It is given by the image of the chain control set
of the original linear control system.
