Stability and Stabilization of Infinite Delay Systems: A Lyapunov-Based Approach

• Published in: IEEE transactions on automatic control Vol. 65; no. 11; pp. 4509 - 4524
• Main Authors: Xu, Xiang, Liu, Lu, Feng, Gang
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.11.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Asymptotic stability; Controllers; Delay; Delay systems; Delays; Frequency-domain analysis; Infinite delay; Linear systems; Lyapunov method; Nonlinear systems; Stability; Stability analysis; Stabilization; System effectiveness; Theorems; Time domain analysis
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2019.2958557

This article addresses stability and stabilization of infinite delay systems, which are more general but also more challenging to deal with than bounded delay systems. Based on a general model of infinite delay systems and a newly approved key technical lemma, several new Lyapunov theorems for uniform asymptotical stability and exponential stability, respectively, are established. The stability results are more general than existing stability results, and the corresponding conditions are more easily satisfied than existing ones. Therefore, the new stability results are expected to be more widely applicable. These new Lyapunov theorems are then applied to the problems of stabilizing both time-invariant and time-varying linear systems with distributed infinite input delays, and the corresponding stabilizing controllers are developed. Key to obtaining the stabilizing controllers is to construct appropriate Lyapunov functionals for these systems via introduction of a regulation term into the Lyapunov functional. A distinctive advantage of the Lyapunov-based time-domain method proposed in this article over the existing frequency domain method is that the former can be adopted to deal with more general systems, such as time-varying linear systems or even nonlinear systems. Several examples are provided to illustrate the effectiveness of our results.