Partial stability and stabilisation of Boolean networks

Abstract

In this paper, we investigate the stability of Boolean networks and the stabilisation of Boolean control networks with respect to part of the system's states. First, an algebraic expression of the Boolean (control) network is derived by the semi-tensor product of matrices. Then, some necessary and sufficient conditions for partial stability of Boolean networks are given. Finally, the stabilisation of Boolean control networks by a free control sequence and a state-feedback control is investigated and the respective necessary and sufficient conditions are obtained. Examples are provided to illustrate the efficiency of the obtained results.

Keywords:

• Boolean (control) network
• partial stability
• partial stabilisation
• state-feedback control
• semi-tensor product of matrices

Acknowledgements

The authors would like to thank the Associate Editor and the reviewers for their constructive comments and suggestions which improved the quality of the paper.

Additional information

Funding

This work was supported by the NSF of Zhejiang Province of China under Grant LY14A010008 and the NNSF of China under Grants 11101373, and 11271333, National Undergraduate Training Programs for Innovation and Entrepreneurship [grant number 201310345010] and the new-shoot Talents Program of Zhejiang province [grant number 2013R404041].

Notes on contributors

Hong-Wei Chen

Hong-Wei Chen received the B.S. degree in Mathematics from Qiongzhou University, Sanya, China, in 2011, and the M.S. degree in Mathematics from Zhejiang Normal University, Jinhua, China, in 2014. He is currently working toward the Ph.D. degree at the Department of Mathematics, Southeast University, Nanjing, China. His research interests include logical dynamic systems, systems biology, etc.

Liang-Jie Sun

Liang-Jie Sun is currently working toward the B.S. degree in Department of Mathematics from Zhejiang Normal University, Jinhua, China. Her research interests include logical dynamic systems, systems biology.

Yang Liu

Yang Liu received the B.S. degree in Mathematics from Zhejiang Normal University, China in 2003, and the Ph.D. degree from Tongji University, Shanghai in 2008. He is currently an Associate Professor in Department of Mathematics, Zhejiang Normal University, China. Previously he was a Research Associate Fellow in Faculty of Science and Technology, University of Macau, Macau, China between January and February, June and July 2014 respectively. He is a Visiting Scholar at Indiana UniversityPurdue University Fort Wayne between August 2014 and August 2015. He was the recipient of the Shanghai Outstanding Ph.D's Thesis Award in 2011. His research interests include several complex analysis, switched systems, Boolean networks.