  Invariance entropy is a measure for the smallest data rate in a noiseless
digital channel above which a controller that only receives state information
through this channel is able to render a given subset of the state space
invariant. In this paper, we derive a lower bound on the invariance entropy for
a class of partially hyperbolic sets. More precisely, we assume that $Q$ is a
compact controlled invariant set of a control-affine system whose extended
tangent bundle decomposes into two invariant subbundles $E^+$ and $E^{0-}$ with
uniform expansion on $E^+$ and weak contraction on $E^{0-}$. Under the
additional assumptions that $Q$ is isolated and that the $u$-fibers of $Q$ vary
lower semicontinuously with the control $u$, we derive a lower bound on the
invariance entropy of $Q$ in terms of relative topological pressure with
respect to the unstable determinant. Under the assumption that this bound is
tight, our result provides a first quantitative explanation for the fact that
the invariance entropy does not only depend on the dynamical complexity on the
set of interest.
