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This paper studies the stabilization and safety problems of nonlineartime-delay systems. Following both Razumikhin and Krasovskii approaches, wepropose novel control Lyapunov functions/functionals for the stabilizationproblem and novel control barrier functions/functionals for the safety problem.The proposed control Lyapunov and barrier functions/functionals extend theexisting ones from the delay-free case to the time-delay case, and allow fordesigning the stabilizing and safety controllers in closed-form. Sinceanalytical solutions to time-delay optimal control problems are hard to beachieved, a sliding mode control based approach is developed to merge theproposed control Lyapunov and barrier functions/functionals. Based on thesliding surface functional, a feedback control law is established toinvestigate the stabilization and safety objectives simultaneously. Inparticular, the properties of the sliding surface functional are analyzed, andfurther how to construct the sliding surface functional is discussed. Finally,the proposed approaches are illustrated via two numerical examples from theconnected cruise control problem of automotive systems and the synchronizationproblem of multi-agent systems.

Razumikhin and Krasovskii Approaches for Safe Stabilization