Input-to-state stability theorem and strict Lyapunov functional constructions for time-delay systems on time scales

ABSTRACT

In this paper, input-to-state stability of nonlinear time-delay systems on time scales is investigated. Due to the advantages of the strict Lyapunov functionals in uncertainty quantification and robustness analysis, one always prefers to construct the strict Lyapunov functionals to analyse stability of time-delay systems. However, it may be not an easy task to do this for some time-delay systems. This paper proposes an input-to-state stability theorem based on a time-scale uniformly asymptotically stable function. The advantage of this theorem is that it is dependent on the non-strict Lyapunov functional, whose time-scale derivative can be non-negative on some time intervals. Then, some approaches are established to construct the strict Lyapunov functionals based on the non-strict ones. It is shown that input-to-state stability theorems can be also formulated in terms of these strict Lyapunov functionals. Finally, to illustrate the effectiveness of the main results, an example is given.

Keywords:

• Input-to-state stability
• time-delay systems
• strict Lyapunov functional
• non-strict Lyapunov functional
• time scale

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [61873150] and the China Postdoctoral Science Foundation [2020M672110].

Notes on contributors

Xiaodong Lu

Xiaodong Lu received the M.S. degree from the School of Mathematical Science, Shandong Normal University in 2015, and received the Ph.D. degree from the School of Control Science and Engineering, Shandong University in 2019. Since 2019, he has been with the School of Mathematical Science, Shandong Normal University, China. His research interests include descriptor systems, systems on time scales and time-delay systems.

Haitao Li

Haitao Li received the B.S. and M.S. degrees from the School of Mathematical Science, Shandong Normal University in 2007 and 2010, respectively, and the Ph.D. degree at the School of Control Science and Engineering, Shandong University in 2014. Since 2015, he has been with the School of Mathematics and Statistics, Shandong Normal University, where he is currently a professor. From Jan. 2014 to Jan. 2015, he worked as a Research Fellow in Nanyang Technological University, Singapore. His research interests include finite-value systems, networked control systems, etc. He received the Young Experts of Taishan Scholar Project in 2019, the Second Class Prize of The Natural Science Award of Shandong Province in 2018, the Distinguished Young Scholars of Shandong Province in 2016, the Guan Zhaozhi Award in 2012 and the Best Student Paper Award at the 10th WCICA in 2012.