  The presence of (partially) automated vehicles on the roads presents an
opportunity to compensate the unstable behaviour of conventional vehicles.
Vehicles subject to perturbations should (i) recover their equilibrium speed,
(ii) react not to propagate but absorb perturbations. In this work, we start
with considering vehicle systems consisting of heterogeneous vehicles updating
their dynamics according to realistic behavioural car-following models.
Definitions of all types of stability that are of interest in the vehicle
system, namely input-output stability, scalability, weak and strict string
stability, are introduced based on recent studies. Then, frequency domain
linear stability analyses are conducted after linearisation of the modelled
system of vehicles, leading to conditions for input-output stability, strict
and weak string stability over the behavioural parameters of the system, for
finite and infinite systems of homogeneous and heterogeneous vehicles. This
provides a solid basis that was missing for car-following model-based control
design in mixed traffic systems where only a proportion of vehicles can be
controlled. After visualisation of the theoretical results in simulation, we
formulate an optimisation strategy with LMI constraints to tune the behavioural
parameters of the automated vehicles in order to maximise the L1 string
stability of the mixed traffic flow while considering the comfort of automated
driving. The optimisation strategy systematically leads to increased traffic
flow stability. We show that very few automated vehicles are required to
prevent the
