Converse Barrier Certificates for Finite-time Safety Verification of Continuous-time Perturbed Deterministic Systems

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.