  This contribution analyzes the widely used and well-known "intelligent driver
model (briefly IDM), which is a second order car-following model governed by a
system of ordinary differential equations. Although this model was intensively
studied in recent years for properly capturing traffic phenomena and driver
braking behavior, a rigorous study of the well-posedness has, to our knowledge,
never been performed. First it is shown that, for a specific class of initial
data, the vehicles' velocities become negative or even diverge to $-\infty$ in
finite time, both undesirable properties for a car-following model. Various
modifications of the IDM are then proposed in order to avoid such
ill-posedness. The theoretical remediation of the model, rather than post facto
by ad-hoc modification of code implementations, allows a more sound numerical
implementation and preservation of the model features. Indeed, to avoid
inconsistencies and ensure dynamics close to the one of the original model, one
may need to inspect and clean large input data, which may result in practically
impossible scenarios for large-scale simulations. Although well-posedness
issues occur only for specific initial data, this may happen frequently when
different traffic scenarios are analyzed, and especially in presence of
lane-changing, on ramps and other network components as it is the case for most
commonly used micro-simulators. On the other side, it is shown that
well-posedness can be guaranteed by straight-forward improvements, such as
those obtained by slightly changing the acceleration to prevent the velocity
from becoming negative.
