  This paper presents conditions for ensuring forward invariance of safe sets
under sampled-data system dynamics with piecewise-constant controllers and
fixed time-steps. First, we introduce two different metrics to compare the
conservativeness of sufficient conditions on forward invariance under
piecewise-constant controllers. Then, we propose three approaches for
guaranteeing forward invariance, two motivated by continuous-time barrier
functions, and one motivated by discrete-time barrier functions. All proposed
conditions are control affine, and thus can be incorporated into quadratic
programs for control synthesis. We show that the proposed conditions are less
conservative than those in earlier studies. The advantages of the new
approaches are demonstrated via simulations on an obstacle-avoidance problem
for a unicycle agent and on a spacecraft attitude reorientation problem, in
which the new approaches achieve the mission objectives for cases where the
existing methods fail.
