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Stability and stabilization of Boolean networks with impulsive effects. (English) Zbl 1250.93078
Summary: This paper investigates the stability and stabilization of Boolean networks with impulsive effects. After giving a survey on semi-tensor product of matrices, we convert a Boolean network with impulsive effects into impulsive discrete-time dynamics. Then, some necessary and sufficient conditions are given for the stability and stabilization of Boolean networks with impulsive effects. Finally, examples are provided to illustrate the efficiency of the obtained results.

93C55 Discrete-time control/observation systems
93D09 Robust stability
92C42 Systems biology, networks
Full Text: DOI
[1] Kauffman, S.A., Metabolic stability and epigenesis in randomly constructed genetic nets, Journal of theoretical biology, 22, 437-467, (1969)
[2] Shmulevich, I.; Dougherty, E.R.; Kim, S.; Zhang, W., Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks, Bioinformatics, 18, 261-274, (2002)
[3] Farrow, C.; Heidel, J.; Maloney, J.; Rogers, J., Scalar equations for synchronous Boolean networks with biological applications, IEEE transactions on neural network, 15, 348-354, (2004)
[4] Albert, R.; Barabási, A.-L., Dynamics of complex systems: scaling laws for the period of Boolean networks, Physical review letters, 84, 5660-5663, (2000)
[5] Aldana, M., Boolean dynamics of networks with scale-free topology, Physica D, 185, 45-66, (2003) · Zbl 1039.94016
[6] Samuelsson, B.; Troein, C., Superpolynomial growth in the number of attractors in kauffman networks, Physical review letters, 90, 1-4, 098701, (2003) · Zbl 1267.82104
[7] Drossel, B.; Mihaljev, T.; Greil, F., Number and length of attractors in a critical kauffman model with connectivity one, Physical review letters, 94, 1-4, 088701, (2005)
[8] Heidel, J.; Maloney, J.; Farrow, C.; Rogers, J.A., Finding cycles in synchronous Boolean networks with applications to biochemical systems, International journal of bifurcation and chaos, 13, 535-552, (2003) · Zbl 1056.37013
[9] Cheng, D., Input-state approach to Boolean networks, IEEE transactions on neural network, 20, 512-521, (2009)
[10] Cheng, D.; Qi, H., A linear representation of dynamics of Boolean networks, IEEE transactions on automatic control, 55, 2251-2258, (2010) · Zbl 1368.37025
[11] Cheng, D.; Qi, H., Controllability and observability of Boolean control networks, Automatica, 45, 1659-1667, (2009) · Zbl 1184.93014
[12] Zhao, Y.; Qi, H.; Cheng, D., Input-state incidence matrix of Boolean control networks and its applications, Systems & control letters, 59, 767-774, (2010) · Zbl 1217.93026
[13] Cheng, D.; Li, Z.; Qi, H., Realization of Boolean control networks, Automatica, 46, 62-69, (2010) · Zbl 1214.93031
[14] Li, F.; Sun, J., Controllability of Boolean control networks with time-delays in states, Automatica, 47, 603-607, (2011) · Zbl 1220.93010
[15] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[16] Yao, J.; Guan, Z.-H.; Chen, G.; Ho, Daniel W.C., Stability, robust stabilization and \(H_\infty\) control of singular-impulsive systems via switching control, Systems & control letters, 55, 879-886, (2006) · Zbl 1113.93096
[17] Hong, Y., Finite-time stabilization and stabilizability of a class of controllable systems, Systems & control letters, 46, 231-236, (2002) · Zbl 0994.93049
[18] Zhang, Y.; Sun, J.; Feng, G., Impulsive control of discrete systems with time delay, IEEE transactions on automatic control, 54, 830-834, (2009)
[19] Carrasco, J.; Banos, A.; van der Schaft, A., A passivity-based approach to reset control systems stability, Systems & control letters, 59, 18-24, (2010) · Zbl 1208.93118
[20] Cheng, D.; Qi, H.; Li, Z.; Liu, J., Stability and stabilization of Boolean networks, International journal of robust and nonlinear control, 21, 134-156, (2011) · Zbl 1213.93121
[21] Huang, S.; Ingber, I., Shape-dependent control of cell growth differentiation, and apotosis: switching between attractors in cell regulatory networks, Experimenatal cell research, 61, 91-103, (2000)
[22] Goodwin, B.C., Temporal organization in cells, (1963), Academic New York
[23] Cheng, D.; Qi, H.; Li, Z., Analysis and control of Boolean networks, (2011), Springer-Verlag Press London
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