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July  2022, 15(7): 1749-1765. doi: 10.3934/dcdss.2022004

State bounding for time-delay impulsive and switching genetic regulatory networks with exogenous disturbance

1. 

School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

2. 

School of Petroleum Engineering, Yangtze University, Jingzhou, Hubei 434023, China

3. 

School of Automation, Central South University, Changsha, Hunan 410000, China

*Corresponding author: Zhi-Hong Guan

Received  August 2021 Revised  November 2021 Published  July 2022 Early access  January 2022

Fund Project: This work was supported in part by the National Natural Science Foundation of China under Grants 61633011, 61772086, 61976100, 61873287, and 61976099

This paper focuses on the state bounding problem for the time-delay impulsive and switching genetic regulatory networks (ISGRNs) with exogenous disturbances. Firstly, a sufficient criterion for the state bounding is obtained such that all the trajectories of ISGRNs under consideration converge exponentially into a sphere on the basis of an average dwell time (ADT) switching. Besides, globally exponential stability conditions for the considered system are further stated when the exogenous disturbance vanishes. As a special case, the equivalent state bounding criteria are established by using the properties of some special matrices when there exist no impulses at the switching instants in ISGRNs. Finally, an illustrating example is given to demonstrate the derived results. Compared with the existing literatures, the considered genetic regulatory networks (GRNs) have more general structure and the approach adopted in the present paper is more simple than Lyapunov-Krasovskii functional (LKF) approach.

Citation: Jiayuan Yan, Ding-Xue Zhang, Bin Hu, Zhi-Hong Guan, Xin-Ming Cheng. State bounding for time-delay impulsive and switching genetic regulatory networks with exogenous disturbance. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1749-1765. doi: 10.3934/dcdss.2022004
References:
[1]

L. Chen and K. Aihara, Stability of genetic regulatory networks with time delay, IEEE Trans. Circuits Systems I Fund. Theory Appl., 49 (2002), 602-608.  doi: 10.1109/TCSI.2002.1001949.

[2]

E. Fridman and U. Shaked, On reachable sets for linear systems with delay and bounded peak inputs, Automatica, 39 (2003), 2005-2010.  doi: 10.1016/S0005-1098(03)00204-8.

[3]

Y. GaoS. ZhuC. Yang and S. Wen, State bounding for fuzzy memristive neural networks with bounded input disturbances, Neural Networks, 134 (2021), 167-172. 

[4]

Z.-H. GuanD. J. Hill and X. Shen, On hybrid impulsive and switching systems and application to nonlinear control, IEEE Trans. Automat. Control, 50 (2005), 1058-1062.  doi: 10.1109/TAC.2005.851462.

[5]

Z. -H. Guan, B. Hu and X. Shen, Introduction to Hybrid Intelligent Networks, Springer, 2019. doi: 10.1007/978-3-030-02161-0.

[6]

Z.-H. GuanD. YueB. HuT. Li and F. Liu, Cluster synchronization of coupled genetic regulatory networks with delays via aperiodically adaptive intermittent control, IEEE Transactions on Nanobioscience, 16 (2017), 585-599. 

[7]

L. V. Hien and H. M. Trinh, A new approach to state bounding for linear time-varying systems with delay and bounded disturbances, Automatica, 50 (2014), 1735-1738.  doi: 10.1016/j.automatica.2014.04.025.

[8]

Y. Ji and X. Liu, Unified synchronization criteria for hybrid switching-impulsive dynamical networks, Circuits Systems and Signal Processing, 34 (2015), 1499-1517.  doi: 10.1007/s00034-014-9916-0.

[9]

B. JiangB. Li and J. Lu, Complex systems with impulsive effects and logical dynamics: A brief overview, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 1273-1299.  doi: 10.3934/dcdss.2020369.

[10]

T. JiaoJ. H. ParkG. ZongJ. Liu and Y. Chen, Stochastic stability analysis of switched genetic regulatory networks without stable subsystems, Appl. Math. Comput., 359 (2019), 261-277.  doi: 10.1016/j.amc.2019.04.059.

[11]

J. LamB. ZhangY. Chen and S. Xu, Reachable set estimation for discrete-time linear systems with time delays, Internat. J. Robust Nonlinear Control, 25 (2015), 269-281.  doi: 10.1002/rnc.3086.

[12]

X. LiP. Li and Q. Wang, Input/output-to-state stability of impulsive switched systems, Systems Control Lett., 116 (2018), 1-7.  doi: 10.1016/j.sysconle.2018.04.001.

[13]

D. Liberzon, Switching in Systems and Control, Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-1-4612-0017-8.

[14]

H. LiuL. GaoZ. Wang and Z. Liu, Asynchronous l2-l filtering of discrete-time impulsive switched systems with admissible edge-dependent average dwell time switching signal, Internat. J. Systems Sci., 52 (2021), 1564-1585.  doi: 10.1080/00207721.2020.1866094.

[15]

X. Liu and P. Stechlinski, Switching and impulsive control algorithms for nonlinear hybrid dynamical systems, Nonlinear Anal. Hybrid Syst., 27 (2018), 307-322.  doi: 10.1016/j.nahs.2017.09.004.

[16]

J. LygerosC. Tomlin and S. Sastry, Controllers for reachability specifications for hybrid systems, Automatica, 35 (1999), 349-370.  doi: 10.1016/S0005-1098(98)00193-9.

[17]

P. T. NamP. N. Pathirana and H. Trinh, Reachable set bounding for nonlinear perturbed time-delay systems: The smallest bound, Appl. Math. Lett., 43 (2015), 68-71.  doi: 10.1016/j.aml.2014.11.015.

[18]

P. H. A. Ngoc and H. Trinh, Novel criteria for exponential stability of linear neutral time-varying differential systems, IEEE Trans. Automat. Control, 61 (2016), 1590-1594.  doi: 10.1109/TAC.2015.2478125.

[19]

S. PandiselviR. RajaJ. Cao and G. Rajchakit, Stabilization of switched stochastic genetic regulatory networks with leakage and impulsive effects, Neural Process Letters, 49 (2019), 593-610. 

[20]

J. QiuK. SunC. YangX. ChenX. Chen and A. Zhang, Finite-time stability of genetic regulatory networks with impulsive effects, Neurocomputing, 219 (2017), 9-14.  doi: 10.1016/j.neucom.2016.09.017.

[21]

H. RenG. ZongL. Hou and Y. Yi, Finite-time control of interconnected impulsive switched systems with time-varying delay, Appl. Math. Comput., 276 (2016), 143-157. 

[22]

T. Shen and I. R. Petersen, An ultimate state bound for a class of linear systems with delay, Automatica, 87 (2018), 447-449.  doi: 10.1016/j.automatica.2017.09.026.

[23]

X. WanZ. WangM. Wu and X. Liu, State estimation for discrete time-delayed genetic regulatory networks with stochastic noises under the round-bobin protocols, IEEE Transactions on Nanobioscience, 17 (2018), 145-154. 

[24]

Z. WangJ. LamG. WeiK. Fraser and X. Liu, Filtering for nonlinear genetic regulatory networks with stochastic disturbances, IEEE Trans. Automat. Control, 53 (2008), 2448-2457.  doi: 10.1109/TAC.2008.2007862.

[25]

W. XiangH. D. Tran and T. T. Johnson, Output reachable set estimation for switched linear systems and its application in safety verification, IEEE Trans. Automat. Control, 62 (2017), 5380-5387.  doi: 10.1109/TAC.2017.2692100.

[26]

Y. XueL. Zhang and X. Zhang, Reachable set estimation for genetic regulatory networks with time-varying delays and bounded disturbances, Neurocomputing, 403 (2020), 203-210.  doi: 10.1016/j.neucom.2020.03.113.

[27]

Y. YangY. LiuJ. Lou and Z. Wang, Observability of switched Boolean control networks using algebraic forms, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 1519-1533.  doi: 10.3934/dcdss.2020373.

[28]

Y. YaoJ. Liang and J. Cao, Stability analysis for switched genetic regulatory networks: An average dwell time approach, J. Franklin Inst., 348 (2011), 2718-2733.  doi: 10.1016/j.jfranklin.2011.04.016.

[29]

T. Yu, Y. Zhao and Q. Zeng, Stability analysis for discrete-time switched GRNs with persistent dwell-time and time delays, J. Franklin Inst., 357 (2020), 11730-11749. doi: 10.1016/j. jfranklin. 2019.09.039.

[30]

D. YueZ.-H. GuanJ. LiF. LiuJ. Xiao and G. Ling, Stability and bifurcation of delay-coupled genetic regulatory networks with hub structure, J. Franklin Inst., 356 (2019), 2847-2869.  doi: 10.1016/j.jfranklin.2018.11.030.

[31]

B. ZhangJ. Lam and S. Xu, Reachable set estimation and controller design for distributed delay systems with bounded disturbances, J. Franklin Inst., 351 (2014), 3068-3088.  doi: 10.1016/j.jfranklin.2014.02.007.

[32]

J. Zhang and Y. Sun, Reachable set estimation for switched nonlinear positive systems with impulse and time delay, Internat. J. Robust Nonlinear Control, 30 (2020), 3332-3343.  doi: 10.1002/rnc.4931.

[33]

N. ZhangY. Sun and F Meng, State bounding for switched homogeneous positive nonlinear systems with exogenous input, Nonlinear Anal. Hybrid Syst., 29 (2018), 363-372.  doi: 10.1016/j.nahs.2018.03.004.

[34]

W. ZhangJ. Fang and W. Cui, Exponential stability of switched genetic regulatory networks with both stable and unstable subsystems, J. Franklin Inst., 350 (2013), 2322-2333.  doi: 10.1016/j.jfranklin.2013.06.007.

[35]

X. Zhang, Y. Wang and L. Wu, Analysis and Design of Delayed Genetic Regulatory Networks, Springer, Cham, 2019. doi: 10.1007/978-3-030-17098-1.

show all references

References:
[1]

L. Chen and K. Aihara, Stability of genetic regulatory networks with time delay, IEEE Trans. Circuits Systems I Fund. Theory Appl., 49 (2002), 602-608.  doi: 10.1109/TCSI.2002.1001949.

[2]

E. Fridman and U. Shaked, On reachable sets for linear systems with delay and bounded peak inputs, Automatica, 39 (2003), 2005-2010.  doi: 10.1016/S0005-1098(03)00204-8.

[3]

Y. GaoS. ZhuC. Yang and S. Wen, State bounding for fuzzy memristive neural networks with bounded input disturbances, Neural Networks, 134 (2021), 167-172. 

[4]

Z.-H. GuanD. J. Hill and X. Shen, On hybrid impulsive and switching systems and application to nonlinear control, IEEE Trans. Automat. Control, 50 (2005), 1058-1062.  doi: 10.1109/TAC.2005.851462.

[5]

Z. -H. Guan, B. Hu and X. Shen, Introduction to Hybrid Intelligent Networks, Springer, 2019. doi: 10.1007/978-3-030-02161-0.

[6]

Z.-H. GuanD. YueB. HuT. Li and F. Liu, Cluster synchronization of coupled genetic regulatory networks with delays via aperiodically adaptive intermittent control, IEEE Transactions on Nanobioscience, 16 (2017), 585-599. 

[7]

L. V. Hien and H. M. Trinh, A new approach to state bounding for linear time-varying systems with delay and bounded disturbances, Automatica, 50 (2014), 1735-1738.  doi: 10.1016/j.automatica.2014.04.025.

[8]

Y. Ji and X. Liu, Unified synchronization criteria for hybrid switching-impulsive dynamical networks, Circuits Systems and Signal Processing, 34 (2015), 1499-1517.  doi: 10.1007/s00034-014-9916-0.

[9]

B. JiangB. Li and J. Lu, Complex systems with impulsive effects and logical dynamics: A brief overview, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 1273-1299.  doi: 10.3934/dcdss.2020369.

[10]

T. JiaoJ. H. ParkG. ZongJ. Liu and Y. Chen, Stochastic stability analysis of switched genetic regulatory networks without stable subsystems, Appl. Math. Comput., 359 (2019), 261-277.  doi: 10.1016/j.amc.2019.04.059.

[11]

J. LamB. ZhangY. Chen and S. Xu, Reachable set estimation for discrete-time linear systems with time delays, Internat. J. Robust Nonlinear Control, 25 (2015), 269-281.  doi: 10.1002/rnc.3086.

[12]

X. LiP. Li and Q. Wang, Input/output-to-state stability of impulsive switched systems, Systems Control Lett., 116 (2018), 1-7.  doi: 10.1016/j.sysconle.2018.04.001.

[13]

D. Liberzon, Switching in Systems and Control, Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-1-4612-0017-8.

[14]

H. LiuL. GaoZ. Wang and Z. Liu, Asynchronous l2-l filtering of discrete-time impulsive switched systems with admissible edge-dependent average dwell time switching signal, Internat. J. Systems Sci., 52 (2021), 1564-1585.  doi: 10.1080/00207721.2020.1866094.

[15]

X. Liu and P. Stechlinski, Switching and impulsive control algorithms for nonlinear hybrid dynamical systems, Nonlinear Anal. Hybrid Syst., 27 (2018), 307-322.  doi: 10.1016/j.nahs.2017.09.004.

[16]

J. LygerosC. Tomlin and S. Sastry, Controllers for reachability specifications for hybrid systems, Automatica, 35 (1999), 349-370.  doi: 10.1016/S0005-1098(98)00193-9.

[17]

P. T. NamP. N. Pathirana and H. Trinh, Reachable set bounding for nonlinear perturbed time-delay systems: The smallest bound, Appl. Math. Lett., 43 (2015), 68-71.  doi: 10.1016/j.aml.2014.11.015.

[18]

P. H. A. Ngoc and H. Trinh, Novel criteria for exponential stability of linear neutral time-varying differential systems, IEEE Trans. Automat. Control, 61 (2016), 1590-1594.  doi: 10.1109/TAC.2015.2478125.

[19]

S. PandiselviR. RajaJ. Cao and G. Rajchakit, Stabilization of switched stochastic genetic regulatory networks with leakage and impulsive effects, Neural Process Letters, 49 (2019), 593-610. 

[20]

J. QiuK. SunC. YangX. ChenX. Chen and A. Zhang, Finite-time stability of genetic regulatory networks with impulsive effects, Neurocomputing, 219 (2017), 9-14.  doi: 10.1016/j.neucom.2016.09.017.

[21]

H. RenG. ZongL. Hou and Y. Yi, Finite-time control of interconnected impulsive switched systems with time-varying delay, Appl. Math. Comput., 276 (2016), 143-157. 

[22]

T. Shen and I. R. Petersen, An ultimate state bound for a class of linear systems with delay, Automatica, 87 (2018), 447-449.  doi: 10.1016/j.automatica.2017.09.026.

[23]

X. WanZ. WangM. Wu and X. Liu, State estimation for discrete time-delayed genetic regulatory networks with stochastic noises under the round-bobin protocols, IEEE Transactions on Nanobioscience, 17 (2018), 145-154. 

[24]

Z. WangJ. LamG. WeiK. Fraser and X. Liu, Filtering for nonlinear genetic regulatory networks with stochastic disturbances, IEEE Trans. Automat. Control, 53 (2008), 2448-2457.  doi: 10.1109/TAC.2008.2007862.

[25]

W. XiangH. D. Tran and T. T. Johnson, Output reachable set estimation for switched linear systems and its application in safety verification, IEEE Trans. Automat. Control, 62 (2017), 5380-5387.  doi: 10.1109/TAC.2017.2692100.

[26]

Y. XueL. Zhang and X. Zhang, Reachable set estimation for genetic regulatory networks with time-varying delays and bounded disturbances, Neurocomputing, 403 (2020), 203-210.  doi: 10.1016/j.neucom.2020.03.113.

[27]

Y. YangY. LiuJ. Lou and Z. Wang, Observability of switched Boolean control networks using algebraic forms, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 1519-1533.  doi: 10.3934/dcdss.2020373.

[28]

Y. YaoJ. Liang and J. Cao, Stability analysis for switched genetic regulatory networks: An average dwell time approach, J. Franklin Inst., 348 (2011), 2718-2733.  doi: 10.1016/j.jfranklin.2011.04.016.

[29]

T. Yu, Y. Zhao and Q. Zeng, Stability analysis for discrete-time switched GRNs with persistent dwell-time and time delays, J. Franklin Inst., 357 (2020), 11730-11749. doi: 10.1016/j. jfranklin. 2019.09.039.

[30]

D. YueZ.-H. GuanJ. LiF. LiuJ. Xiao and G. Ling, Stability and bifurcation of delay-coupled genetic regulatory networks with hub structure, J. Franklin Inst., 356 (2019), 2847-2869.  doi: 10.1016/j.jfranklin.2018.11.030.

[31]

B. ZhangJ. Lam and S. Xu, Reachable set estimation and controller design for distributed delay systems with bounded disturbances, J. Franklin Inst., 351 (2014), 3068-3088.  doi: 10.1016/j.jfranklin.2014.02.007.

[32]

J. Zhang and Y. Sun, Reachable set estimation for switched nonlinear positive systems with impulse and time delay, Internat. J. Robust Nonlinear Control, 30 (2020), 3332-3343.  doi: 10.1002/rnc.4931.

[33]

N. ZhangY. Sun and F Meng, State bounding for switched homogeneous positive nonlinear systems with exogenous input, Nonlinear Anal. Hybrid Syst., 29 (2018), 363-372.  doi: 10.1016/j.nahs.2018.03.004.

[34]

W. ZhangJ. Fang and W. Cui, Exponential stability of switched genetic regulatory networks with both stable and unstable subsystems, J. Franklin Inst., 350 (2013), 2322-2333.  doi: 10.1016/j.jfranklin.2013.06.007.

[35]

X. Zhang, Y. Wang and L. Wu, Analysis and Design of Delayed Genetic Regulatory Networks, Springer, Cham, 2019. doi: 10.1007/978-3-030-17098-1.

Figure 1.  (a) Switching sequence with ADT switching in the case 1 and (b) Switching sequence with ADT switching in the case 2
Figure 2.  (a) State trajectories $ X_1,X_2,X_3 $ in the case 1 and (b) State trajectories $ Y_1,Y_2,Y_3 $ in the case 1
Figure 3.  (a) State trajectories $ X_1,X_2,X_3 $ in the case 2 and (b) State trajectories $ Y_1,Y_2,Y_3 $ in the case 2
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