A New Slack Lyapunov Functional for Dynamical System with Time Delay

• Published in: Mathematics (Basel) Vol. 10; no. 23; p. 4462
• Main Authors: Zhao, Can, Shi, Kaibo, Tang, Yiqian, Zhong, Shouming
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01-12-2022
• Subjects: Analysis; asymptotic stability; Delay lines; dynamical system; Dynamical systems; inequality; Liapunov functions; Mathematical analysis; Matrices (mathematics); new Lyapunov functional; Quadratic equations; Quadratic forms; Time lag; time-varying delays; China
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math10234462

The traditional method of constructing a Lyapunov functional for dynamical systems with time delay is usually dependent on positive definite matrices in the quadratic form. In this paper, a new Lyapunov functional is proposed and the corresponding proof is given. It do not require that all matrices in the quadratic form of Lyapunov functionals are positive definite, while the quadratic form is still positive definite, which makes the estimate more relaxed due to special construction of matrices. It is a general form and can be used in the performance analysis of a variety of dynamical systems. Moreover, a lemma concerning the quadratic function is applied to deal with the quadratic term of time-varying delay. Lastly, in the case of classical dynamical systems with time delay, the verification results are given to illustrate the usefulness of the new slack Lyapunov functional.