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  • Xian ZhangEmail author
  • Yantao Wang
  • Ligang Wu
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 207)

Abstract

In this chapter we will briefly introduce some background knowledge related to Genetic Regulatory Networks (GRNs).

References

  1. 1.
    Ahmad, F.K., Deris, S., Othman, N.H.: The inference of breast cancer metastasis through gene regulatory networks. J. Biomed. Inf. 45(2), 350–362 (2012)CrossRefGoogle Scholar
  2. 2.
    Akutsu, T., Miyano, S., Kuhara, S., et al.: Identification of genetic networks from a small number of gene expression patterns under the Boolean network model. In: Pacific Symposium on Biocomputing, vol. 4, pp. 17–28. World Scientific Maui, Hawaii (1999)Google Scholar
  3. 3.
    Amato, F., Ariola, M., Dorato, P.: Finite-time control of linear systems subject to parametric uncertainties and disturbances. Automatica 37(9), 1459–1463 (2001)zbMATHCrossRefGoogle Scholar
  4. 4.
    Balasubramaniam, P., Sathy, R.: Robust asymptotic stability of fuzzy Markovian jumping genetic regulatory networks with time-varying delays by delay decomposition approach. Commun. Nonlinear Sci. Numer. Simul. 16(2), 928–939 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Banu, L.J., Balasubramaniam, P.: Non-fragile observer design for discrete-time genetic regulatory networks with randomly occurring uncertainties. Phys. Scr. 90 (Article ID 015205, 2015)CrossRefGoogle Scholar
  6. 6.
    Beal, M.J., Falciani, F., Ghahramani, Z., Rangel, C., Wild, D.L.: A Bayesian approach to reconstructing genetic regulatory networks with hidden factors. Bioinformatics 21(3), 349–356 (2005)CrossRefGoogle Scholar
  7. 7.
    Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory, vol. 15. SIAM, Philadelphia, PA (1994)Google Scholar
  8. 8.
    Britton, N.F.: Reaction-Diffusion Equations and Their Applications to Biology. Academic Press, New York (1986)Google Scholar
  9. 9.
    Busenberg, S., Mahaffy, J.: Interaction of spatial diffusion and delays in models of genetic control by repression. J. Mol. Biol. 22(3), 313–333 (1985)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Butte, A.J., Kohane, I.S.: Mutual information relevance networks: functional genomic clustering using pairwise entropy measurements. In: Pacific Symposium on Biocomputing, vol. 5, pp. 415–426 (2000)Google Scholar
  11. 11.
    Cao, J., Ren, F.: Exponential stability of discrete-time genetic regulatory networks with delays. IEEE Trans. Neural Netw. 19(3), 520–523 (2008)CrossRefGoogle Scholar
  12. 12.
    Chen, L., Aihara, K.: Stability of genetic regulatory networks with time delay. IEEE Trans. Circuits Syst. I: Regul. Pap. 49(5), 602–608 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Chen, L., Zhao, S., Zhu, W., Liu, Y., Zhang, W.: A self-adaptive differential evolution algorithm for parameters identification of stochastic genetic regulatory system with random delays. Arab. J. Sci. Eng. 39(2), 821–835 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Chen, L.L., Zhou, Y., Zhang, X.: Guaranteed cost control for uncertain genetic regulatory networks with interval time-varying delays. Neurocomputing 131, 105–112 (2014)CrossRefGoogle Scholar
  15. 15.
    Chen, T., He, H.L., Church, G.M.: Modeling gene expression with differential equations. Pac. Symp. Biocomput. 4, 29–40 (1999)Google Scholar
  16. 16.
    Darabos, C., Di Cunto, F., Tomassini, M., Moore, J., Provero, P., Giacobini, M.: Additive functions in Boolean models of gene regulatory network modules. PloS One 6(11) (Article ID e25110, 2011)CrossRefGoogle Scholar
  17. 17.
    De Jong, H.: Modeling and simulation of genetic regulatory systems: a literature review. J. Comput. Biol. 9(1), 67–103 (2002)CrossRefGoogle Scholar
  18. 18.
    de Oliveira, M.C., Bernussou, J., Geromel, J.C.: A new discrete-time robust stability condition. Syst. Control Lett. 37(4), 261–265 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Elowitz, M.B., Leibler, S.: A synthetic oscillatory network of transcriptional regulators. Nature 403(20), 335–338 (2000)CrossRefGoogle Scholar
  20. 20.
    Friedman, N.: Inferring cellular networks using probabilistic graphical models. Science 303(5659), 799–805 (2004)CrossRefGoogle Scholar
  21. 21.
    Golub, T.R., Slonim, D.K., Tamayo, P., Huard, C., Gaasenbeek, M., Mesirov, J.P., Coller, H., Loh, M.L., Downing, J.R., Caligiuri, M.A., et al.: Molecular classification of cancer: class discovery and class prediction by gene expression monitoring. Science 286(5439), 531–537 (1999)CrossRefGoogle Scholar
  22. 22.
    Graudenzi, A., Serra, R., Villani, M., Damiani, C., Colacci, A., Kauffman, S.: Dynamical properties of a Boolean model of gene regulatory network with memory. J. Comput. Biol. 18(10), 1291–1303 (2011)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Gu, K.: A further refinement of discretized Lyapunov functional method for the stability of time-delay systems. Int. J. Control 74(10), 967–976 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Han, Y., Zhang, X., Wang, Y.: Asymptotic stability criteria for genetic regulatory networks with time-varying delays and reaction-diffusion terms. Circuits Syst. Sign. Process. 34(10), 3161–3190 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Han, Y.Y., Zhang, X.: Stability analysis for delayed regulatory networks with reaction-diffusion terms (in Chinese). J. Nat. Sci. Heilongjiang Univ. 31(1), 32–40 (2014)zbMATHGoogle Scholar
  26. 26.
    He, W., Cao, J.: Robust stability of genetic regulatory networks with distributed delay. Cogn. Neurodyn. 2(4), 355–361 (2008)CrossRefGoogle Scholar
  27. 27.
    He, Y., Zeng, J., Wu, M., Zhang, C.K.: Robust stabilization and \(H_{\infty }\) controllers design for stochastic genetic regulatory networks with time-varying delays and structured uncertainties. Math. Biosci. 236(1), 53–63 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Hirata, H., Yoshiura, S., Ohtsuka, T., Bessho, Y., Harada, T., Yoshik awa, K., Kageyama, R.: Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop. Science 298, 840–843 (2002)CrossRefGoogle Scholar
  29. 29.
    Hong, Y.G., Xu, Y.S., Huang, J.: Finite-time control for robot manipulators. Syst. Control Lett. 46(4), 243–253 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Hu, J.Q., Liang, J.L., Cao, J.D.: Stabilization of genetic regulatory networks with mixed time-delays: an adaptive control approach. IMA J. Math. Control Inf. 32(2), 343–358 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Johnstone, R.W., Ruefli, A.A., Lowe, S.W.: Apoptosis: a link between cancer genetics and chemotherapy. Cell 108(2), 153–164 (2002)CrossRefGoogle Scholar
  32. 32.
    Kalir, S., Mangan, S., Alon, U.: A coherent feed-forward loop with a SUM input function prolongs flagella expression in Escherichia coli. Mol. Syst. Biol. 1 (Article No. 2005.0006, 2005)Google Scholar
  33. 33.
    Kaluza, P., Inoue, M.: Design of artificial genetic regulatory networks with multiple delayed adaptive responses. Eur. Phys. J. B 89(6) (Article ID 156, 2016)Google Scholar
  34. 34.
    Koo, J.H., Ji, D.H., Won, S.C., Park, J.H.: An improved robust delay-dependent stability criterion for genetic regulatory networks with interval time delays. Commun. Nonlinear Sci. Numer. Simul. 17, 3339–3405 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Krstić, M., Deng, H.: Stabilization of Nonlinear Uncertain Systems. Springer-Verlag, London (1998)Google Scholar
  36. 36.
    Li, C., Chen, L., Aihara, K.: Stability of genetic networks with SUM regulatory logic: Lur’e system and LMI approach. IEEE Trans. Circuits Syst. I: Regul. Pap. 53(11), 2451–2458 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Li, L., Yang, Y.Q.: On sampled-data control for stabilization of genetic regulatory networks with leakage delays. Neurocomputing 149, 1225–1231 (2015)CrossRefGoogle Scholar
  38. 38.
    Liu, J.L., Yue, D.: Asymptotic and robust stability of T-S fuzzy genetic regulatory networks with time-varying delays. Int. J. Robust Nonlinear Control 22(8), 827–840 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Liu, T.T., Zhang, X., Gao, X.Y.: Stability analysis for continuous-time and discrete-time genetic regulatory networks with delays. Appl. Math. Comput. 274, 628–643 (2016)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Liu, Z., Jiang, H.: Exponential stability of genetic regulatory networks with mixed delays by periodically intermittent control. Neural Comput. Appl. 21(6), 1263–1269 (2012)CrossRefGoogle Scholar
  41. 41.
    Luo, Q., Zhang, R., Liao, X.: Unconditional global exponential stability in Lagrange sense of genetic regulatory networks with SUM regulatory logic. Cogn. Neurodyn. 4(3), 251–261 (2010)CrossRefGoogle Scholar
  42. 42.
    Ma, C., Zeng, Q., Zhang, L., Zhu, Y.: Passivity and passification for Markov jump genetic regulatory networks with time-varying delays. Neurocomputing 136, 321–326 (2014)CrossRefGoogle Scholar
  43. 43.
    Ma, Q., Shi, G.D., Xu, S.Y., Zou, Y.: Stability analysis for delayed genetic regulatory networks with reaction-diffusion terms. Neural Comput. Appl. 20(4), 507–516 (2011)CrossRefGoogle Scholar
  44. 44.
    Mohammadian, M., Momeni, H.R., Karimi, H.S., Shafikhani, I., Tahmasebi, M.: An LPV based robust peak-to-peak state estimation for genetic regulatory networks with time varying delay. Neurocomputing 160, 261–273 (2015)CrossRefGoogle Scholar
  45. 45.
    Monk, N.A.M.: Oscillatory expression of Hes1, p53, and NF-\(\kappa \)B driven by transcriptional time delays. Current Biol. 13(16), 1409–1413 (2003)CrossRefGoogle Scholar
  46. 46.
    Norbury, J., Stuart, A.M.: Volterra integral equations and a new Gronwall inequality (Part I: the linear case). Proc. R. Soc. Edinb.: Section A Math. 106(3–4), 361–373 (1987)zbMATHCrossRefGoogle Scholar
  47. 47.
    Park, P., Lee, W.I., Lee, S.Y.: Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems. J. Franklin Inst. 352(4), 1378–1396 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Pease, A.C., Solas, D., Sullivan, E.J., Cronin, M.T., Holmes, C.P., Fodor, S.: Light-generated oligonucleotide arrays for rapid DNA sequence analysis. Proc. Natl. Acad. Sci. 91(11), 5022–5026 (1994)CrossRefGoogle Scholar
  49. 49.
    Plemmons, R.J.: M-matrix characterizations. I–nonsingular M-matrices. Linear Algebra Appl. 18(2), 175–188 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Qian, W., Cong, S., Li, T., Fei, S.: Improved stability conditions for systems with interval time-varying delay. Int. J. Control Autom. Syst. 10(6), 1146–1152 (2012)CrossRefGoogle Scholar
  51. 51.
    Rakkiyappan, R., Balasubramaniam, P.: Delay-probability-distribution-dependent stability of uncertain stochastic genetic regulatory networks with mixed time-varying delays: an LMI approach. Nonlinear Anal.: Hybrid Syst. 4(3), 600–607 (2010)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Ren, F., Cao, J.: Asymptotic and robust stability of genetic regulatory networks with time-varying delays. Neurocomputing 71(4), 834–842 (2008)CrossRefGoogle Scholar
  53. 53.
    Ren, F.L., Cao, F., Cao, J.D.: Mittag-leffler stability and generalized mittag-leffler stability of fractional-order gene regulatory networks. Neurocomputing 160, 185–190 (2015)CrossRefGoogle Scholar
  54. 54.
    Saadatpour, A., Albert, R.: Boolean modeling of biological regulatory networks: a methodology tutorial. Methods 62(1), 3–12 (2013)CrossRefGoogle Scholar
  55. 55.
    Salimpour, A., Sojoodi, M., Majd, V.: Robust stability analysis of stochastic genetic regulatory networks with discrete and distributed delay in both mRNA and protein dynamics. In: Proceedings of the 2010 IEEE Conference on Cybernetics and Intelligent Systems (CIS), pp. 7–13. IEEE (2010)Google Scholar
  56. 56.
    Seuret, A., Gouaisbaut, F., Fridman, E.: Stability of systems with fast-varying delay using improved Wirtinger’s inequality. In: Proceedings of the 52nd IEEE Annual Conference on Decision and Control (CDC), pp. 946–951. IEEE (2013)Google Scholar
  57. 57.
    Shen, H., Huang, X., Zhou, J., Wang, Z.: Global exponential estimates for uncertain Markovian jump neural networks with reaction-diffusion terms. Nonlinear Dyn. 69(1–2), 473–486 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Smart, D.R.: Fixed Point Theorems, Cambridge Tracts in Mathematics, vol. 66. Cambridge University Press, Cambridge (1980)Google Scholar
  59. 59.
    Somogyi, R., Sniegoski, C.: Modeling the complexity of genetic networks: understanding multigenic and pleiotropic regulation. Complexity 1, 45–63 (1996)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Strauss, W.A.: Partial Differential Equations: An Introduction. Wiley, New York (1992)Google Scholar
  61. 61.
    Sun, J., Liu, G.P., Chen, J.: Delay-dependent stability and stabilization of neutral time-delay systems. Int. J. Robust Nonlinear Control 19(12), 1364–1375 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Tang, Y., Wang, Z., Fang, J.A.: Parameters identification of unknown delayed genetic regulatory networks by a switching particle swarm optimization algorithm. Expert Syst. Appl. 38(3), 2523–2535 (2011)CrossRefGoogle Scholar
  63. 63.
    Thieffry, D., Thomas, R.: Qualitative analysis of gene networks. In: Pacific Symposium on Biocomputing, vol. 3, pp. 77–88 (1998)Google Scholar
  64. 64.
    Tian, L.P., Shi, Z.K., Liu, L.Z., Wu, F.X.: M-matrix-based stability conditions for genetic regulatory networks with time-varying delays and noise perturbations. IET Syst. Biol. 7(5), 214–222 (2013)CrossRefGoogle Scholar
  65. 65.
    Tian, L.P., Shi, Z.K., Wu, F.X.: New global stability conditions for genetic regulatory networks with time-varying delays. In: Proceedings of 2012 IEEE 6th International Conference on Systems Biology (ISB), pp. 185–191. IEEE (2012)Google Scholar
  66. 66.
    Tian, L.P., Wu, F.X.: Globally delay-independent stability of ring-structured genetic regulatory networks. In: Proceedings of the 24th Canadian Conference on Electrical and Computer Engineering (CCECE), pp. 000308–000311 (2011)Google Scholar
  67. 67.
    Wan, X.B., Xu, L., Fang, H.J., Ling, G.: Robust non-fragile \(H_{\infty }\) state estimation for discrete-time genetic regulatory networks with Markov jump delays and uncertain transition probabilities. Neurocomputing 154, 162–173 (2015)CrossRefGoogle Scholar
  68. 68.
    Wang, W., Zhong, S.: Delay-dependent stability criteria for genetic regulatory networks with time-varying delays and nonlinear disturbance. Commun. Nonlinear Sci. Numer. Simul. 17, 3597–3611 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Wang, W., Zhong, S.: Stochastic stability analysis of uncertain genetic regulatory networks with mixed time-varying delays. Neurocomputing 82, 143–156 (2012)CrossRefGoogle Scholar
  70. 70.
    Wang, W., Zhong, S., Liu, F., Cheng, J.: Robust delay-probability-distribution-dependent stability of uncertain stochastic genetic regulatory networks with random discrete delays and distributed delays. Int. J. Robust Nonlinear Control 24(16), 2574–2596 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Wang, W.Q., Zhong, S.M., Liu, F.: New delay-dependent stability criteria for uncertain genetic regulatory networks with time-varying delays. Neurocomputing 93, 19–26 (2012)CrossRefGoogle Scholar
  72. 72.
    Wang, Y., Xie, L., de Souza, C.E.: Robust control of a class of uncertain nonlinear systems. Syst. Control Lett. 19(2), 139–149 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Wang, Y.T., Yu, A.H., Zhang, X.: Robust stability of stochastic genetic regulatory networks with time-varying delays: a delay fractioning approach. Neural Comput. Appl. 23(5), 1217–1227 (2013)CrossRefGoogle Scholar
  74. 74.
    Wang, Y.T., Zhang, X., Hu, Z.R.: Delay-dependent robust \(H_{\infty }\) filtering of uncertain stochastic genetic regulatory networks with mixed time-varying delays. Neurocomputing 166, 346–356 (2015)CrossRefGoogle Scholar
  75. 75.
    Wang, Y.T., Zhou, X.M., Zhang, X.: \(H_{\infty }\) filtering for discrete-time genetic regulatory networks with random delay described by a Markovian chain. Abstr. Appl. Anal. 2014 (Article ID 257971, 12 pages, 2014)Google Scholar
  76. 76.
    Weaver, D.C., Workman, C.T., Stormo, G.D.: Modeling regulatory networks with weight matrices. In: Pacific Symposium on Biocomputing, vol. 4, pp. 112–123. World Scientific Maui, Hawaii (1999)Google Scholar
  77. 77.
    Wu, F.X.: Delay-independent stability of genetic regulatory networks. IEEE Trans. Neural Netw. 22(11), 1685–1693 (2011)CrossRefGoogle Scholar
  78. 78.
    Wu, F.X.: Global and robust stability analysis of genetic regulatory networks with time-varying delays and parameter uncertainties. IEEE Trans. Biomed. Circuits Syst. 5(4), 391–398 (2011)CrossRefGoogle Scholar
  79. 79.
    Wu, F.X.: Stability and bifurcation of ring-structured genetic regulatory networks with time delays. IEEE Trans. Circuits Syst. I: Regul. Pap. 59(6), 1312–1320 (2012)MathSciNetCrossRefGoogle Scholar
  80. 80.
    Wu, H., Liao, X., Guo, S., Feng, W., Wang, Z.: Stochastic stability for uncertain genetic regulatory networks with interval time-varying delays. Neurocomputing 72(13–15), 3263–3276 (2009)CrossRefGoogle Scholar
  81. 81.
    Wu, H., Liao, X.F., Wei, F., Guo, S.T., Zhang, W.: Robust stability for uncertain genetic regulatory networks with interval time-varying delays. Inf. Sci. 180(18), 3532–3545 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    Xiao, M., Cao, J.D.: Genetic oscillation deduced from Hopf bifurcation in a genetic regulatory network with delays. Math. Biosci. 215(1), 55–63 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Yan, R., Liu, J.: New results on asymptotic and robust stability of genetic regulatory networks with time-varying delays. Int. J. Innov. Comput. Inf. Control 8(4), 2889–2900 (2012)Google Scholar
  84. 84.
    Yeh, H.Y., Cheng, S.W., Lin, Y.C., Yeh, C.Y., Lin, S.F., Soo, V.W.: Identifying significant genetic regulatory networks in the prostate cancer from microarray data based on transcription factor analysis and conditional independency. BMC Med. Genomics 2 (Article ID 70, 2009)Google Scholar
  85. 85.
    Yu, T.T., Zhang, X., Zhang, G.D., Niu, B.: Hopf bifurcation analysis for genetic regulatory networks with two delays. Neurocomputing 164, 190–200 (2015)CrossRefGoogle Scholar
  86. 86.
    Zang, H., Zhang, T., Zhang, Y.: Bifurcation analysis of a mathematical model for genetic regulatory networks with time delays. Appl. Math. Comput. 260, 204–226 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  87. 87.
    Zhang, W., Fang, J., Tang, Y.: Stochastic stability of Markovian jumping genetic regulatory networks with mixed time delays. Appl. Math. Comput. 217(17), 7210–7225 (2011)MathSciNetzbMATHGoogle Scholar
  88. 88.
    Zhang, W.B., Fang, J.A., Tang, Y.: New robust stability analysis for genetic regulatory networks with random discrete delays and distributed delays. Neurocomputing 74(14–15), 2344–2360 (2011)CrossRefGoogle Scholar
  89. 89.
    Zhang, X., Han, Y., Wu, L., Wang, Y.: State estimation for delayed genetic regulatory networks with reaction-diffusion terms. IEEE Trans. Neural Netw. Learn. Syst. 29(2), 299–309 (2018)MathSciNetCrossRefGoogle Scholar
  90. 90.
    Zhang, X., Han, Y.Y., Wu, L., Zou, J.H.: M-matrix-based globally asymptotic stability criteria for genetic regulatory networks with time-varying discrete and unbounded distributed delays. Neurocomputing 174, 1060–1069 (2016)CrossRefGoogle Scholar
  91. 91.
    Zhang, X., Wu, L., Cui, S.C.: An improved integral inequality to stability analysis of genetic regulatory networks with interval time-varying delays. IEEE/ACM Trans. Comput. Biol. Bioinf. 12(2), 398–409 (2015)CrossRefGoogle Scholar
  92. 92.
    Zhang, X., Wu, L., Zou, J.H.: Globally asymptotic stability analysis for genetic regulatory networks with mixed delays: an M-matrix-based approach. IEEE/ACM Trans. Comput. Biol. Bioinf. 13(1), 135–147 (2016)CrossRefGoogle Scholar
  93. 93.
    Zhang, X., Yu, A.H., Zhang, G.D.: M-matrix-based delay-range-dependent global asymptotical stability criterion for genetic regulatory networks with time-varying delays. Neurocomputing 113, 8–15 (2013)CrossRefGoogle Scholar
  94. 94.
    Zhang, Y., Liu, H., Yan, F., Zhou, J.: Oscillatory behaviors in genetic regulatory networks mediated by microrna with time delays and reaction-diffusion terms. IEEE Trans. Nanobiosci. 16(3), 166–176 (2017)CrossRefGoogle Scholar
  95. 95.
    Zhou, J.P., Xu, S.Y., Shen, H.: Finite-time robust stochastic stability of uncertain stochastic delayed reaction-diffusion genetic regulatory networks. Neurocomputing 74(17), 2790–2796 (2011)CrossRefGoogle Scholar
  96. 96.
    Zhou, Q., Shao, X.Y., Karimi, H.R., Zhu, J.: Stability of genetic regulatory networks with time-varying delay: Delta operator method. Neurocomputing 149, 490–495 (2015)CrossRefGoogle Scholar
  97. 97.
    Zhou, Q., Xu, S.Y., Chen, B., Li, H.Y., Chu, Y.M.: Stability analysis of delayed genetic regulatory networks with stochastic disturbances. Phys. Lett. A 373(41), 3715–3723 (2009)zbMATHCrossRefGoogle Scholar
  98. 98.
    Zhu, Y., Zhang, Q., Wei, Z., Zhang, L.: Robust stability analysis of Markov jump standard genetic regulatory networks with mixed time delays and uncertainties. Neurocomputing 110, 44–50 (2013)CrossRefGoogle Scholar
  99. 99.
    Zhu, Z., Zhu, Y., Zhang, L., Al-Yami, M., Abouelmagd, E., Ahmad, B.: Mode-mismatched estimator design for Markov jump genetic regulatory networks with random time delays. Neurocomputing 168, 1121–1131 (2015)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.School of Mathematical ScienceHeilongjiang UniversityHarbinChina
  2. 2.School of AstronauticsHarbin Institute of TechnologyHarbinChina

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