Skip to main content
SpringerLink
Log in

Asymptotic Stability Criteria for Genetic Regulatory Networks with Time-Varying Delays and Reaction–Diffusion Terms

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

This paper investigates the asymptotic stability problem for delayed genetic regulatory networks with reaction–diffusion terms under both Dirichlet boundary conditions and Neumann boundary conditions. First, by constructing a new Lyapunov–Krasovskii functional and using Jensen’s inequality, Wirtinger’s inequality, Green’s second identity and the reciprocally convex approach, we establish delay-dependent asymptotic stability criteria that do not require a restriction of the upper bounds of the delays’ derivatives being less than 1. Thus, the stability criteria that we establish are less conservative than the existing criteria and extend the range of applications of the theoretical results. In addition, it is shown that the obtained criterion under Dirichlet boundary conditions retains the information about the reaction–diffusion terms, while these do not exist in the criterion under Neumann boundary conditions. It is then theoretically presented that the stability criteria established in this paper are less conservative than the existing ones. Finally, numerical examples are given to illustrate the effectiveness of the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. T. Akutsu, S. Miyano, S. Kuhara, 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 (World Scientific Maui, Hawaii 1999), pp. 17–28

  2. P. Balasubramaniam, G. Nagamani, A delay decomposition approach to delay-dependent robust passive control for Takagi–Sugeno fuzzy nonlinear systems. Circuits Syst. Signal Process. 31(4), 1319–1341 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. M.J. Beal, F. Falciani, Z. Ghahramani, C. Rangel, D.L. Wild, A Bayesian approach to reconstructing genetic regulatory networks with hidden factors. Bioinformatics 21(3), 349–356 (2005)

    Article  Google Scholar 

  4. N.F. Britton et al., Reaction–Diffusion Equations and Their Applications to Biology (Academic Press, New York, 1986)

    MATH  Google Scholar 

  5. S. Busenberg, J. Mahaffy, Interaction of spatial diffusion and delays in models of genetic control by repression. J. Math. Biol. 22(3), 313–333 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. B. Chen, L. Yu, W.A. Zhang, \(H_\infty \) filtering for Markovian switching genetic regulatory networks with time-delays and stochastic disturbances. Circuits Syst. Signal Process. 30(6), 1231–1252 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. H. De Jong, Modeling and simulation of genetic regulatory systems: a literature review. J. Comput. Biol. 9(1), 67–103 (2002)

    Article  Google Scholar 

  8. Y. Du, Y. Li, R. Xu, Stability analysis for impulsive stochastic reaction–diffusion differential system and its application to neural networks. J. Appl. Math. 2013, 12 (2013). doi:10.1155/2013/785141

  9. K. Gu, An integral inequality in the stability problem of time-delay systems, in Proceedings of the 39th IEEE Conference on Decision and Control (2000), pp. 2805–2810

  10. Y. Han, X. Zhang, Stability analysis for delayed regulatory networks with reaction-diffusion terms (in Chinese). J. Nat. Sci. Heilongjiang Univ. 31(1), 32–40 (2014)

    Google Scholar 

  11. S. He, F. Liu, On delay-dependent stability of Markov jump systems with distributed time-delays. Circuits Syst. Signal Process. 30(2), 323–337 (2011)

    Article  MATH  Google Scholar 

  12. M. Hua, F. Deng, X. Liu, Y. Peng, Robust delay-dependent exponential stability of uncertain stochastic system with time-varying delay. Circuits Syst. Signal Process. 29(3), 515–526 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Huang, Z. Xiang, Delay-dependent robust \(H_{\infty }\) control for 2-D discrete nonlinear systems with state delays. Multidimens. Syst. Signal Process. 25(4), 775–794 (2013)

    Article  Google Scholar 

  14. S. Huang, Z. Xiang, Delay-dependent stability for discrete 2-D switched systems with state delays in the Roesser model. Circuits Syst. Signal Process. 32(6), 2821–2837 (2013)

    Article  MathSciNet  Google Scholar 

  15. S. Huang, Z. Xiang, H.R. Karimi, Input-output finite-time stability of discrete-time impulsive switched linear systems with state delays. Circuits Syst. Signal Process. 33(1), 141–158 (2014)

    Article  MathSciNet  Google Scholar 

  16. T.C. Lacalli, Modeling the Drosophila pair-rule pattern by reaction–diffusion: gap input and pattern control in a 4-morphogen system. J. Theor. Biol. 144(2), 171–194 (1990)

    Article  MathSciNet  Google Scholar 

  17. F. Li, L. Wu, P. Shi, Stochastic stability of semi-Markovian jump systems with mode-dependent delays. Int. J. Robust Nonlinear Control 24(18), 3317–3330 (2014)

    Article  MathSciNet  Google Scholar 

  18. H. Li, H. Gao, P. Shi, New passivity analysis for neural networks with discrete and distributed delays. IEEE Trans. Neural Netw. 21(11), 1842–1847 (2010)

    Article  Google Scholar 

  19. Y. Li, Y. Zhu, N. Zeng, M. Du, Stability analysis of standard genetic regulatory networks with time-varying delays and stochastic perturbations. Neurocomputing 74(17), 3235–3241 (2011)

    Article  Google Scholar 

  20. X. Lin, X. Zhang, Y.T. Wang, Robust passive filtering for neutral-type neural networks with time-varying discrete and unbounded distributed delays. J. Franklin Inst. 350(5), 966–989 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Löfberg, YAPLMI: a toolbox for modeling and optimization in MATLAB, in Proceedings of the 2004 IEEE International Symposium on Computer Aided Control Systems Design (CACSD), (2004) pp. 284–289

  22. Q. Luo, R. Zhang, X. Liao, Unconditional global exponential stability in Lagrange sense of genetic regulatory networks with SUM regulatory logic. Cogn. Neurodynamics 4(3), 251–261 (2010)

    Article  Google Scholar 

  23. Q. Ma, G.D. Shi, S.Y. Xu, Y. Zou, Stability analysis for delayed genetic regulatory networks with reaction–diffusion terms. Neural Comput. Appl. 20(4), 507–516 (2011)

    Article  Google Scholar 

  24. S. Muralisankar, N. Gopalakrishnan, P. Balasubramaniam, Robust exponential stability criteria for T-S fuzzy stochastic delayed neural networks of neutral type. Circuits Syst. Signal Process. 30(6), 1617–1641 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. P.G. Park, J.W. Ko, C. Jeong, Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47(1), 235–238 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. J. Qiu, G. Feng, H. Gao, Observer-based piecewise affine output feedback controller synthesis of continuous-time T-S fuzzy affine dynamic systems using quantized measurements. IEEE Trans. Fuzzy Syst. 20(6), 1046–1062 (2012)

    Article  Google Scholar 

  27. J. Qiu, G. Feng, H. Gao, Static-output-feedback control of continuous-time T-S fuzzy affine systems via piecewise Lyapunov functions. IEEE Trans. Fuzzy Syst. 21(2), 245–261 (2013)

    Article  Google Scholar 

  28. J. Qiu, G. Feng, J. Yang, A new design of delay-dependent robust filtering for discrete-time T-S fuzzy systems with time-varying delay. IEEE Trans. Fuzzy Syst. 17(5), 1044–1058 (2009)

    Article  Google Scholar 

  29. R. Rakkiyappan, S. Lakshmanan, P. Balasubramaniam, Delay-probability-distribution-dependent stability of uncertain stochastic genetic regulatory networks with time-varying delays. Circuits Syst. Signal Process. 32, 1147–1177 (2013)

    Article  MathSciNet  Google Scholar 

  30. A. Saadatpour, R. Albert, Boolean modeling of biological regulatory networks: a methodology tutorial. Methods 62(1), 3–12 (2013)

    Article  Google Scholar 

  31. H. Shen, X. Huang, J. Zhou, Z. Wang, Global exponential estimates for uncertain Markovian jump neural networks with reaction–diffusion terms. Nonlinear Dyn. 69(1–2), 473–486 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. P. Shi, X. Luan, C.L. Liu, Filtering for discrete-time systems with stochastic incomplete measurement and mixed delays. IEEE Trans. Ind. Electron. 59(6), 2732–2739 (2012)

    Article  Google Scholar 

  33. P. Shi, Y. Zhang, R.K. Agarwal, Stochastic finite-time state estimation for discrete time-delay neural networks with Markovian jumps. Neurocomputing 151(1), 168–174 (2015)

    Article  Google Scholar 

  34. W.A. Strauss, Partial Differential Equations: An Introduction (Wiley, New York, 1992)

    MATH  Google Scholar 

  35. J.F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11(1–4), 625–653 (1999)

    Article  MathSciNet  Google Scholar 

  36. X. Su, P. Shi, L. Wu, Y.D. Song, A novel control design on discrete-time Takagi–Sugeno fuzzy systems with time-varying delays. IEEE Trans. Fuzzy Syst. 21(4), 655–671 (2013)

    Article  Google Scholar 

  37. Y.H. Sun, G. Feng, J. Cao, Robust stochastic stability analysis of genetic regulatory networks with disturbance attenuation. Neurocomputing 79, 39–49 (2012)

    Article  Google Scholar 

  38. T. Tian, K. Burragea, P.M. Burragea, M. Carlettib, Stochastic delay differential equations for genetic regulatory networks. Int. J. Comput. Math. 205, 696–707 (2007)

    MATH  Google Scholar 

  39. J. Wang, S. Liu, The stability analysis of a general viral infection model with distributed delays and multi-staged infected progression. Commun. Nonlinear Sci. Numer. Simul. 20(1), 263–272 (2015)

    Article  MathSciNet  Google Scholar 

  40. J. Wang, J. Pang, T. Kuniya, Y. Enatsu, Global threshold dynamics in a five-dimensional virus model with cell-mediated, humoral immune responses and distributed delays. Appl. Math. Comput. 241, 298–316 (2014)

    Article  MathSciNet  Google Scholar 

  41. Y.T. Wang, X. Zhang, Y. He, Improved delay-dependent robust stability criteria for a class of uncertain mixed neutral and Lur’e dynamical systems with interval time-varying delays and sector-bounded nonlinearity. Nonlinear Anal. Real World Appl. 13(5), 2188–2194 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Y.T. Wang, X. Zhang, Y.M. Hu, Robust \(H_{\infty }\) control for a class of uncertain neutral stochastic systems with mixed delays: a CCL approach. Circuits Syst. Signal Process. 32(2), 631–646 (2013)

    Article  MathSciNet  Google Scholar 

  43. H. Wu, X.F. Liao, F. Wei, S.T. Guo, W. Zhang, Robust stability for uncertain genetic regulatory networks with interval time-varying delays. Inf. Sci. 180(18), 3532–3545 (2010)

    Article  MATH  Google Scholar 

  44. G.D. Zhang, X. Lin, X. Zhang, Exponential stabilization of neutral-type neural networks with mixed interval time-varying delays by intermittent control: a CCL approach. Circuits Syst. Signal Process. 33, 371–391 (2014). doi:10.1007/s00034-013-9651-y

    Article  Google Scholar 

  45. X. Zhang, L. Wu, S. Cui, An improved integral inequality to stability analysis of genetic regulatory networks with interval time-varying delays. IEEE/ACM Trans. Comput. Biol. Bioinf. 19(13), 1493–1518 (2015)

  46. J. Zhou, S. Xu, H. Shen, Finite-time robust stochastic stability of uncertain stochastic delayed reaction–diffusion genetic regulatory networks. Neurocomputing 74(17), 2790–2796 (2011)

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (11371006), the National Natural Science Foundation of Heilongjiang Province (F201326, A201416), the fund of Heilongjiang Province Innovation Team Support Plan (2012TD007), the Fund of Key Laboratory of Electronics Engineering, College of Heilongjiang Province,(Heilongjiang University), P.R. China, and the Fund of Heilongjiang Education Committee (12541603). The authors thank the anonymous referees for their helpful comments and suggestions, which greatly improved this paper. The authors also thank Dr. Jinliang Wang at Heilongjiang University for checking for grammatical and compositional errors in the previous version.

Author information

Authors and Affiliations

  1. School of Mathematical Science, Heilongjiang University, Harbin, 150080, China

    Yuanyuan Han, Xian Zhang & Yantao Wang

Authors
  1. Yuanyuan Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xian Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yantao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xian Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Han, Y., Zhang, X. & Wang, Y. Asymptotic Stability Criteria for Genetic Regulatory Networks with Time-Varying Delays and Reaction–Diffusion Terms. Circuits Syst Signal Process 34, 3161–3190 (2015). https://doi.org/10.1007/s00034-015-0006-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-015-0006-8

Keywords

Search

Navigation