A Semiglobal Approach for Stabilizing Nonlinear Systems With State Delays by Memoryless Linear Feedback

Abstract

This article investigates the problem of how to control nonlinear systems with delays in the state by memoryless linear feedback. The notions of semiglobal asymptotic stabilization and sublevel sets are introduced in the context of time-delay systems. With the aid of Razumikhin theorem, we develop a semiglobal design method for the construction of Lyapunov functions, associated sublevel sets, and delay-free linear state feedback laws, step-by-step, achieving semiglobal asymptotic stabilization for time-delay nonlinear systems in a lower triangular form. In contrast to the global stabilization of nonlinear systems with delays in the state, which is usually achieved by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dynamic state</i> feedback (Lin and Zhang, 2020), the significance of this work is to point out that a tradeoff of the control objectives, e.g., semiglobal versus global stabilization, makes it possible to control a class of time-delay nonlinear systems by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">static linear</i> feedback. Extensions to nonlinear systems with globally asymptotically locally expoentially stable (GALES)-like inverse dynamics are also included in this article.