Non-monotonicity of Lyapunov functions for functional differential equations with enlightenments for related research methods

Abstract

In this paper, a particular property of Lyapunov functions for functional differential equations (FDEs) is developed, that is the direct dependence of the signs of the derivatives of the Lyapunov functions on the initial data. This property implies that the derivatives of the Lyapunov functions for FDEs cannot be guaranteed to be negative definite generally, and then makes the FDEs differ from the ordinary differential equations constitutionally. With this property, we give some enlightenments for the research methods for establishing stability theorems or criteria for FDEs, which may help us to form a common view about the choice of the investigation methods on the stability of FDEs. The conclusion is stated in both the deterministic and stochastic versions. Two illustrative examples are given to show and verify our conclusion through the paper.

Keywords:

• functional differential equation
• stability
• Lyapunov function
• Halanay inequality
• Razumikhin technique

Acknowledgements

The authors gratefully acknowledge the editor, the associate editor and reviewer's comments.

Additional information

Funding

The authors would like to thank the National Natural Science Foundation of China [grant number 61273126]; the Research Fund for the Doctoral Program of Higher Education of China [grant number 20130172110027]; the Fundamental Research Funds for the Central Universities [grant number 2013ZZ0056] for their financial support.

Notes on contributors

Xueyan Zhao

Xueyan Zhao was born in 1984. She received her PhD degree in systems engineering from South China University of Technology, Guangzhou, in June 2014. Now she is a teacher at South China University of Technology. Her research interests are stability and stabilization of stochastic systems and nonlinear systems.

Feiqi Deng

Feiqi Deng was born in 1962. He received his PhD degree in control theory and control engineering from South China University of Technology, Guangzhou, in June 1997. Since October 1999, he has been a professor with South China University of Technology and the director of the Systems Engineering Institute of the university. He is currently a member of Technical Committee on Control Theory (TCCT), Chinese Association of Automation, and now he is serving as the chair of the IEEE SMC Guangzhou Chapter, an associate editor-in-chief of Journal of South China University of Technology, and a member of the editorial boards of the following journals: Control Theory and Applications, All about Systems and Control, Journal of Systems Engineering and Electronics, and Journal of Systems Engineering, etc. His main research interests include stability, stabilization, and robust control theory of complex systems, including time-delay systems, non-linear systems and stochastic systems. E-mail: [email protected].