  This paper deals with the problem of string stability in a chain of
acceleration-controlled vehicles, i.e. how input disturbances affect the
distributed system for very long chains. There exist variants of string
stability, like avoiding that a local disturbance gets amplified along the
chain, or more strongly ensuring that the output vector's p-norm remains
bounded for any bounded vector of input disturbances independently of the
string length. They are all impossible to achieve with any linear controller if
the vehicles only use relative information of few vehicles in front. Previous
work has shown that adding absolute velocity into the controller, allows to at
least avoid amplification of a local disturbance. In this paper, we consider
the stronger definitions of string stability, under this same relaxation of
using absolute velocity in the controller. We prove that the influence from
input vector to output vector cannot be bounded independently of chain length
in the most popular 2-norm sense, with any bounded stabilizing linear
controller; while a proportional derivative (PD) controller can guarantee it in
the practically relevant infinity-norm sense. Moreover, we identify the
disturbance acting on the leader as the main issue for string stability.
