  Safety Index Synthesis (SIS) is critical for deriving safe control laws.
Recent works propose to synthesize a safety index (SI) via nonlinear
programming and derive a safe control law such that the system 1) achieves
forward invariant (FI) with some safe set and 2) guarantees finite time
convergence (FTC) to that safe set. However, real-world system dynamics can
vary during run-time, making the control law infeasible and invalidating the
initial SI. Since the full SIS nonlinear programming is computationally
expensive, it is infeasible to re-synthesize the SI each time the dynamics are
perturbed. To address that, this paper proposes an efficient approach to
adapting the SI to varying system dynamics and maintaining the feasibility of
the safe control law. The proposed method leverages determinant gradient ascend
and derives a closed-form update to safety index parameters, enabling real-time
adaptation performance. A numerical study validates the effectiveness of our
approach.
