Exponential stability of coupled systems on networks with mixed delays and reaction-diffusion terms. (English) Zbl 1474.35369

Li, Wenxue; Chen, Tianrui; Wang, Ke

Abstr. Appl. Anal. 2014, Article ID 780387, 9 p. (2014).

Summary: This paper is concerned with the stability analysis issue for coupled systems on networks with mixed delays and reaction-diffusion terms (CSNMRs). By employing Lyapunov method and Kirchhoff’s Theorem in graph theory, a systematic method is proposed to guarantee exponential stability of CSNMRs. Two different kinds of sufficient criteria are derived in the form of Lyapunov function and coefficients of the system, respectively. Finally, a numerical example is given to show the effectiveness of the proposed criteria.

MSC:

35K51	Initial-boundary value problems for second-order parabolic systems
35B35	Stability in context of PDEs
35K57	Reaction-diffusion equations

References:

[1]	Strogatz, S. H., Exploring complex networks, Nature, 410, 6825, 268-276 (2001) · Zbl 1370.90052
[2]	Boccaletti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D.-U., Complex networks: structure and dynamics, Physics Reports, 424, 4-5, 175-308 (2006) · Zbl 1371.82002
[3]	He, W.; Cao, J., Exponential synchronization of hybrid coupled networks with delayed coupling, IEEE Transactions on Neural Networks, 21, 4, 571-583 (2010)
[4]	Yu, W.; Cao, J.; Lu, W., Synchronization control of switched linearly coupled neural networks with delay, Neurocomputing, 73, 4-6, 858-866 (2010)
[5]	Zhu, Q.; Cao, J., Stability analysis for stochastic neural networks of neutral type with both Markovian jump parameters and mixed time delays, Neurocomputing, 73, 13-15, 2671-2680 (2010)
[6]	Meng, Q.; Jiang, H., Robust stochastic stability analysis of Markovian switching genetic regulatory networks with discrete and distributed delays, Neurocomputing, 74, 1-3, 362-368 (2010)
[7]	Zhu, Q.; Huang, C.; Yang, X., Exponential stability for stochastic jumping BAM neural networks with time-varying and distributed delays, Nonlinear Analysis: Hybrid Systems, 5, 1, 52-77 (2011) · Zbl 1216.34081
[8]	Gan, Q.; Xu, R., Stability and Hopf bifurcation of a delayed reaction-diffusion neural network, Mathematical Methods in the Applied Sciences, 34, 12, 1450-1459 (2011) · Zbl 1228.35041
[9]	Liu, Y.; Wang, Z.; Liu, X., Stability analysis for a class of neutral-type neural networks with Markovian jumping parameters and mode-dependent mixed delays, Neurocomputing, 94, 46-53 (2012)
[10]	Hu, C.; Jiang, H.; Teng, Z., Globally exponential stability for delayed neural networks under impulsive control, Neural Processing Letters, 31, 2, 105-127 (2010)
[11]	Yu, W.; Cao, J.; Lü, J., Global synchronization of linearly hybrid coupled networks with time-varying delay, SIAM Journal on Applied Dynamical Systems, 7, 1, 108-133 (2008) · Zbl 1161.94011
[12]	Hua, M.; Liu, X.; Deng, F.; Fei, J., New results on robust exponential stability of uncertain stochastic neural networks with mixed time-varying delays, Neural Processing Letters, 32, 3, 219-233 (2010)
[13]	Gan, Q., Exponential synchronization of stochastic Cohen-Grossberg neural networks with mixed time-varying delays and reaction-diffusion via periodically intermittent control, Neural Networks, 31, 12-21 (2012) · Zbl 1245.93125
[14]	Liu, Y.; Jiang, H., Exponential stability of genetic regulatory networks with mixed delays by periodically intermittent control, Neural Computing and Applications, 21, 6, 1263-1269 (2012)
[15]	Gan, Q., Exponential synchronization of stochastic fuzzy cellular neural networks with reaction-diffusion terms via periodically intermittent control, Neural Processing Letters, 37, 393-410 (2013)
[16]	Zhu, Q.; Cao, J., Exponential stability analysis of stochastic reaction-diffusion Cohen-Grossberg neural networks with mixed delays, Neurocomputing, 74, 17, 3084-3091 (2011)
[17]	Yu, F.; Jiang, H., Global exponential synchronization of fuzzy cellular neural networks with delays and reaction-diffusion terms, Neurocomputing, 74, 4, 509-515 (2011)
[18]	Zhu, Q.; Song, B., Exponential stability of impulsive nonlinear stochastic differential equations with mixed delays, Nonlinear Analysis: Real World Applications, 12, 5, 2851-2860 (2011) · Zbl 1223.93102
[19]	Zhu, Q.; Cao, J., Stability of Markovian jump neural networks with impulse control and time varying delays, Nonlinear Analysis: Real World Applications, 13, 5, 2259-2270 (2012) · Zbl 1254.93157
[20]	Rodrigues, M. A.; Odloak, D., New robust stable MPC using linear matrix inequalities, Brazilian Journal of Chemical Engineering, 17, 1, 51-69 (2000)
[21]	Lee, D.; Joo, Y.; Tak, M., Linear matrix inequality approach to local stability analysis of discrete-time Takagi-Sugeno fuzzy systems, IET Control Theory and Applications, 7, 1309-1318 (2013)
[22]	Wang, H.; Song, Q.; Duan, C., LMI criteria on exponential stability of BAM neural networks with both time-varying delays and general activation functions, Mathematics and Computers in Simulation, 81, 4, 837-850 (2010) · Zbl 1204.92006
[23]	Li, M. Y.; Shuai, Z., Global-stability problem for coupled systems of differential equations on networks, Journal of Differential Equations, 248, 1, 1-20 (2010) · Zbl 1190.34063
[24]	Guo, H.; Li, M. Y.; Shuai, Z., A graph-theoretic approach to the method of global Lyapunov functions, Proceedings of the American Mathematical Society, 136, 8, 2793-2802 (2008) · Zbl 1155.34028
[25]	Guo, H.; Li, M.; Shuai, Z., Global dynamics of a general class of multistage models for infectious diseases, SIAM Journal on Applied Mathematics, 72, 261-279 (2012) · Zbl 1244.34075
[26]	Li, M. Y.; Shuai, Z.; Wang, C., Global stability of multi-group epidemic models with distributed delays, Journal of Mathematical Analysis and Applications, 361, 1, 38-47 (2010) · Zbl 1175.92046
[27]	Shu, H.; Fan, D.; Wei, J., Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Analysis: Real World Applications, 13, 4, 1581-1592 (2012) · Zbl 1254.92085
[28]	Zhang, C.; Li, W.; Wang, K., Boundedness for network of stochastic coupled van der Pol oscillators with time-varying delayed coupling, Applied Mathematical Modelling, 37, 5394-5402 (2013) · Zbl 1438.34236
[29]	Zhang, C.; Li, W.; Su, H.; Wang, K., A graph-theoretic approach to boundedness of stochasic Cohen-Grossberg neural networks with Markovian switching, Applied Mathematics and Computation, 219, 9165-9173 (2013) · Zbl 1311.60065
[30]	Su, H.; Li, W.; Wang, K., Global stability analysis of discrete-time coupled systems on networks and its applications, Chaos, 22 (2012)
[31]	Li, W.; Su, H.; Wei, D.; Wang, K., Global stability of coupled nonlinear systems with Markovian switching, Communications in Nonlinear Science and Numerical Simulation, 17, 6, 2609-2616 (2012) · Zbl 1248.93169
[32]	Li, W.; Su, H.; Wang, K., Global stability analysis for stochastic coupled systems on networks, Automatica, 47, 1, 215-220 (2011) · Zbl 1209.93158
[33]	Li, W.; Song, H.; Qu, Y.; Wang, K., Global exponential stability for stochastic coupled systems on networks with Markovian switching, Systems and Control Letters, 62, 468-474 (2013) · Zbl 1279.93104
[34]	Li, W.; Pang, L.; Su, H.; Wang, K., Global stability for discrete Cohen-Grossberg neural networks with finite and infinite delays, Applied Mathematics Letters, 25, 2246-2251 (2012) · Zbl 1262.39021
[35]	Gan, Q., Adaptive synchronization of stochastic neural networks with mixed time delays and reaction-diffusion terms, Nonlinear Dynamics, 69, 4, 2207-2219 (2012) · Zbl 1263.35143
[36]	Gan, Q., Global exponential synchronization of generalized stochastic neural networks with mixed time-varying delays and reaction-diffusion terms, Neurocomputing, 89, 96-105 (2012)
[37]	Su, H.; Qu, Y.; Gao, S.; Song, H.; Wang, K., A model of feedback control system on network and its stability analysis, Communications in Nonlinear Science and Numerical Simulation, 18, 7, 1822-1831 (2013) · Zbl 1273.93079