Safety Without Initial History Assumption

Background: The successful application of Control Barrier Functions (CBFs) to time-delay systems requires ensuring safety conditions are met over the entire time-delay interval, often by relying on assumptions about the initial input history.

Question / Future Work: Developing methods for safety-critical control with input delay that do not require the strong Assumption 3, which mandates that the initial input history over the entire delay interval $[0, \tau]$ must already satisfy the safety conditions.