  In this paper, we show that under mild controllability assumptions a
time-invariant Control Barrier Function (CBF) can be constructed based on
predictions with a finite horizon. As a starting point, we require only a known
subset of a control-invariant set where the latter set does not need to be
explicitly known. We show that, based on ideas similar to the Hamilton-Jacobi
reachability analysis, the knowledge on the subset of a control-invariant set
allows us to obtain a time-invariant CBF for the time-invariant dynamics under
consideration. We also provide a thorough analysis of the properties of the
constructed CBF, we characterize the impact of the prediction horizon, and
comment on the practical implementation. In the end, we relate our construction
approach to Model Predictive Control (MPC). With a relevant application
example, we demonstrate how our method is applied.
