  In this paper, we investigate the problem of verifying the finite-time safety
of continuous-time perturbed deterministic systems represented by ordinary
differential equations in the presence of measurable disturbances. Given a
finite time horizon, if the system is safe, it, starting from a compact initial
set, will remain within an open and bounded safe region throughout the
specified time horizon, regardless of the disturbances. The main contribution
of this work is to uncover that there exists a time-dependent barrier
certificate if and only if the system is safe. This barrier certificate
satisfies the following conditions: negativity over the initial set at the
initial time instant, non-negativity over the boundary of the safe set, and
non-increasing behavior along the system dynamics over the specified finite
time horizon. The existence problem is explored using a Hamilton-Jacobi
differential equation, which has a unique Lipschitz viscosity solution.
