A Lyapunov--Krasovskii Method for Semiglobal Stabilization of a Class of Time-Delay Nonlinear Systems With Uncontrollable Linearization by Linear Feedback

This article presents a seemingly surprising yet simple solution to the open problem of when and how time-delay nonlinear systems with uncontrollable linearization is stabilizable by linear feedback. We show that if a less aggressive control objective, namely, semiglobal asymptotic stabilization (SGAS) instead of GAS is sought, instantaneous linear state feedback would be sufficient for the control of a class of nonlinear systems with state delays, despite the lack of local exponential stabilizability. The SGAS result is obtained by developing a novel Lyapunov--Krasovskii functional framework for the recursive construction of a set of Lyapunov--Krasovskii functionals, associated sublevel sets in Banach space, and linear state controllers systematically, which can be viewed as a semiglobal backstepping method linking Lyapunov functionals to sublevel sets in Banach space.