  In kinetic theory, a system is usually described by its one-particle
distribution function $f(\mathbf{r},\mathbf{v},t)$, such that
$f(\mathbf{r},\mathbf{v},t)d\mathbf{r} d\mathbf{v}$ is the fraction of
particles with positions and velocities in the intervals $(\mathbf{r},
\mathbf{r}+d\mathbf{r})$ and $(\mathbf{v}, \mathbf{v}+d\mathbf{v})$,
respectively. Therein, global stability and the possible existence of an
associated Lyapunov function or $H$-theorem are open problems when
non-conservative interactions are present, as in granular fluids. Here, we
address this issue in the framework of a lattice model for granular-like
velocity fields. For a quite general driving mechanism, including both boundary
and bulk driving, we show that the steady state reached by the system in the
long time limit is globally stable. This is done by proving analytically that a
certain $H$-functional is non-increasing in the long time limit. Moreover, for
two specific energy injection mechanisms, we are able to demonstrate that the
proposed $H$-functional is non-increasing for all times. Also, we put forward a
proof that clearly illustrates why the "classical" Boltzmann functional
$H_{B}[f]=\int\! d\mathbf{r} \, d\mathbf{v} f(\mathbf{r},\mathbf{v},t) \ln
f(\mathbf{r},\mathbf{v},t)$ is inadequate for systems with non-conservative
interactions. Not only is this done for the simplified kinetic description that
holds in the lattice models analysed here but also for a general kinetic
equation, like Boltzmann's or Enskog's.
