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Mathematical Problems in Engineering
Volume 2017, Article ID 4157256, 16 pages
Research Article

Stability Analysis of Delayed Genetic Regulatory Networks via a Relaxed Double Integral Inequality

1The Office of Science and Technology Development, Peking University, Beijing 100871, China
2The Energy Research Institute, State Grid Corporation of China, Beijing, China
3School of Electronic Engineering, Xi’an Shiyou University, Xi’an 710065, China
4Department of Electrical Engineering & Electronics, University of Liverpool, Liverpool L69 3GJ, UK

Correspondence should be addressed to Qi Zhu; nc.ude.uysx@uhzq

Received 29 June 2017; Accepted 12 October 2017; Published 13 November 2017

Academic Editor: Radek Matušů

Copyright © 2017 Fu-Dong Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Time delay arising in a genetic regulatory network may cause the instability. This paper is concerned with the stability analysis of genetic regulatory networks with interval time-varying delays. Firstly, a relaxed double integral inequality, named as Wirtinger-type double integral inequality (WTDII), is established to estimate the double integral term appearing in the derivative of Lyapunov-Krasovskii functional with a triple integral term. And it is proved theoretically that the proposed WTDII is tighter than the widely used Jensen-based double inequality and the recently developed Wiringter-based double inequality. Then, by applying the WTDII to the stability analysis of a delayed genetic regulatory network, together with the usage of useful information of regulatory functions, several delay-range- and delay-rate-dependent (or delay-rate-independent) criteria are derived in terms of linear matrix inequalities. Finally, an example is carried out to verify the effectiveness of the proposed method and also to show the advantages of the established stability criteria through the comparison with some literature.