Finite-time boundedness of stochastic nonlinear reaction-diffusion systems with time delays and exogenous disturbances via boundary control

Document Type : Research Paper

Authors

Department of Mathematics, SRM Institute of Science and Technology, Ramapuram Campus, Chennai- 600 089, Tamil Nadu, India

Abstract

This paper investigates an finite-time boundedness of stochastic nonlinear reaction-diffusion systems (SNRDSs) with time delays and exogenous disturbances via boundary control. Both SNRDSs with and without exogenous disturbances are discussed. We design boundary controller is efforts to achieve the required dynamic behaviors of such SNRDSs. By utilizing Lyapunov-Krasovskii functional (LKF), Wirtinger’s inequality, Gronwall inequality, and linear matrix inequalities (LMIs), sufficient conditions are derived to guarantee the finite-time bounded of proposed systems. Furthermore, the control gain matrices are defined for desired boundary controller. At last, two numerical examples are provided to demonstrate the efficacy and validity of obtained main results.

Keywords

1] M.S. Ali, M. Hymavathi, G. Rajchakit, S. Saroha, L. Palanisamy and P. Hammachukiattikul, Synchronization of
fractional order fuzzy BAM neural networks with time-varying delays and reaction-diffusion terms, IEEE Access
8 (2020), 186551–186571.
[2] M.S. Ali, L. Palanisamy, J. Yogambigai and L. Wang, Passivity-based synchronization of Markovian jump complex
dynamical networks with time-varying delays, parameter uncertainties, reaction-diffusion terms, and sampled-data
control, J. Comput. Appl. Math. 352 (2019), 79–92.
[3] M.S. Ali, S. Saravanan and J. Cao, Finite-time boundedness, L2-gain analysis and control of Markovian jump
switched neural networks with additive time-varying delays, Nonlinear Anal. Hybrid Syst. 23 (2017), 27–43.
[4] M.S. Ali, J. Yogambigai and O.M. Kwon, Finite-time robust passive control for a class of switched reactiondiffusion stochastic complex dynamical networks with coupling delays and impulsive control, Int. J. Syst. Sci. 49
(2018), 718–735.
[5] P. Balasubramaniam and C. Vidhya, Global asymptotic stability of stochastic BAM neural networks with distributed delays and reaction-diffusion terms, J. Comput. Appl. Math. 234 (2010), 3458–3466.
[6] T. Chen, S. Peng, Y. Hong and G. Mai, Finite-time stability and stabilization of impulsive stochastic delayed
neural networks with Rous and Rons, IEEE Access 8 (2020), 87133–87141.
[7] P. Cheng, F. Deng and F. Yao, Almost sure exponential stability and stochastic stabilization of stochastic differential systems with impulsive effects, Nonlinear Anal. Hybrid Syst. 30 (2018), 106–117.
[8] D. Ding, Z. Wang, B. Shen and G. Wei, Event-triggered consensus control for discrete-time stochastic multi-agent
systems: the input-to-state stability in probability, Automatica 62 (2015), 284–291.
[9] T. Dong, A. Wang, H. Zhu and X. Liao, Event-triggered synchronization for reaction-diffusion complex networks
via random sampling, Phys. A: Stat. Mech. Appl. 495 (2018), 454–462.
[10] A.M. Elaiw, A.D. Hobiny and A.D. Al Agha, Global dynamics of reaction-diffusion oncolytic M1 virotherapy with
immune response, Appl. Math. Comput. 367 (2020).
[11] M. Eshaghi Gordji and H. Habibi, Existence and uniqueness of solutions to a first-order differential equation via
fixed point theorem in orthogonal metric space, Facta Univ. Ser. Math. Inf. 34 (2019), 123–135.
[12] M. Eshaghi Gordji, B. Hayati, M. Kamyar and H. Khodaei, On stability and nonstability of systems of functional
equations, Quaest. Math. 44 (2021), 557–567.
[13] N. Espitia, I. Karafyllis and M. Krstic, Event-triggered boundary control of constant-parameter reaction-diffusion
PDEs: A small-gain approach, Automatica 128 (2021).
[14] X. Fan, X. Zhang, L. Wu and M. Shi, Finite-time stability analysis of reaction-diffusion genetic regulatory networks
with time-varying delays, IEEE/ACM Trans. Comput. Biol. Bioinf. 14 (2017), 868–879.
[15] X.X. Han, K.N. Wu and X. Ding, Finite-time stabilization for stochastic reaction-diffusion systems with Markovian
switching via boundary control, Appl. Math. Comput. 385 (2020), 125422.
[16] X.X. Han, K.N. Wu, X. Ding and B. Yang, Boundary control of stochastic reaction-diffusion systems with Markovian switching, Int. J. Robust Nonlinear Control. 30 (2020), 4129–4148.
[17] S. Lakshmanan, M. Prakash, R. Rakkiyappan and J. H. Young, Adaptive synchronization of reaction-diffusion
neural networks and its application to secure communication, IEEE Trans. Cybern. 50 (2020), 911–922.
[18] H. Lin and F. Wang, Global dynamics of a nonlocal reaction-diffusion system modeling the west nile virus transmission, Nonlinear Anal. Real World Appl. 46 (2019), 352–373.
[19] X.Z. Liu, K.N. Wu and Z.T. Li, Exponential stabilization of reaction-diffusion systems via intermittent boundary
control, IEEE Trans. Autom. Control 2021 (2021).
[20] X.Z. Liu, K.N. Wu and W. Zhang, Mean square finite-time boundary stabilisation and H∞ boundary control for
stochastic reaction-diffusion systems, Int. J. Syst. Sci. 50 (2019), 1388–1398.
[21] X.Z. Liu, K.N. Wu, X. Ding and W. Zhang, Boundary stabilization of stochastic delayed Cohen-Grossberg neuralnetworks with diffusion terms, IEEE Trans. Neural Netw. Learn. Syst. 2021 (2021), 1–11.
[22] X.Z. Liu, K.N. Wu and W. Zhang, Intermittent boundary stabilization of stochastic reaction-diffusion CohenGrossberg neural networks, Neural Netw. 131 (2020), 1–13.
[23] G. Narayanan, M. Syed Ali, M. Irshad Alam, G. Rajchakit, N. Boonsatit, P. Kumar and P. Hammachukiattikul, Adaptive fuzzy feedback controller design for finite-time Mittag-Leffler synchronization of fractional-order
quaternion-valued reaction-diffusion fuzzy molecular modeling of delayed neural networks, IEEE Access 9 (2021),
130862–130883.
[24] S. Pandiselvi, R. Raja, J. Cao and G. Rajchakit, Stabilization of switched stochastic genetic regulatory networks
with leakage and impulsive effects, Neural Process. Lett. 49 (2019), 593–610.
[25] A. Pratap, R. Raja, R.P. Agarwal, J. Alzabut, M. Niezabitowski and E. Hincal, Further results on asymptotic
and finite-time stability analysis of fractional-order time-delayed genetic regulatory networks, Neurocomput. 475
(2022), 26–37.
[26] B. Priya, M. Syed Ali, G.K. Thakur, S. Sanober and B. Dhupia, pth moment exponential stability of memristor
Cohen-Grossberg BAM neural networks with time-varying delays and reaction-diffusion, Chinese J. Phys. 74
(2021), 184–194.
[27] G. Rajchakit and R. Sriraman, Robust passivity and stability analysis of uncertain complex-valued impulsive neural
networks with time-varying delays, Neural Process. Lett. 53 (2021), 581–606.
[28] R. Samidurai, R. Manivannan, C. Ki Ahn and H.R. Karimi, New criteria for stability of generalized neural
networks including Markov jump parameters and additive time delays, IEEE Trans. Syst. Man. Cybern. Syst. 48
(2018), 485–499.
[29] R. Samidurai, R. Sriraman, Robust dissipativity analysis for uncertain neural networks with additive time-varying
delays and general activation functions, Math Comput. Simul. 155 (2019), 201–216.
[30] X. Song, J. Man, S. Song, Y. Zhang and Z. Ning, Finite/fixed-time synchronization for Markovian complex-valued
memristive neural networks with reaction-diffusion terms and its application, Neurocomput. 414 (2020), 131–142.
[31] X. Song, M. Wang, J.H. Park and S. Song, Spatial-L∞-norm-based finite-time bounded control for semilinear
parabolic PDE systems with applications to chemical-reaction processes, IEEE Trans. Cybern. 52 (2020), no. 1,
178–191.
[32] X. Song, Q. Zhang, S. Song and C.K. Ahn, Sampled-data-based event-triggered fuzzy control for PDE systems
under cyber-attacks, IEEE Trans. Fuzzy Syst. doi: 10.1109/TFUZZ.2021.3092200.
[33] G.K. Thakur, M. Syed Ali, B. Priya, V. Gokulakrishnan and S. Asma Kauser, Impulsive effects on stochastic
bidirectional associative memory neural networks with reaction-diffusion and leakage delays, Int. J. Comput. Math.
to appear, https://doi.org/10.1080/00207160.2021.1999428.
[34] R. Vadivel and Y. Hoon Joo, Reliable fuzzy H∞-control for permanent magnet synchronous motor against stochastic actuator faults, IEEE Trans. Syst., Man, Cybern. Syst. 51 (2019), 2232–2245.
[35] J. Wang and H. Wu, Passivity of delayed reaction-diffusion networks with application to a food web model, Appl.
Math. Comput. 219 (2013), 11311–11326.
[36] J.L. Wang, X. Zhang, H. Wu, T. Huang and Q. Wang, Finite-time passivity and synchronization of coupled
reaction-diffusion neural networks with multiple weights, IEEE Trans. Cybern. 49 (2019), 3385–3397.
[37] K.N. Wu, M.Y. Na, L. Wang, X. Ding and B. Wu, Finite-time stability of impulsive reaction-diffusion systems
with and without time delay, Appl. Math. Comput. 363 (2019).
[38] R. Wei, J. Cao and J. Kurths, Fixed-time output synchronization of coupled reaction-diffusion neural networks
with delayed output couplings, IEEE Trans. Netw. Sci. Eng. 8 (2021), 780–789.
[39] T. Wei, L. Wang and Y. Wang, Existence, uniqueness and stability of mild solutions to stochastic reaction-diffusion
Cohen-Grossberg neural networks with delays and Wiener processes, Neurocomput. 239 (2017), 19–27.
[40] K. Wu, H. Sun, P. Shi and C.C. Lim, Finite-time boundary stabilization of reaction-diffusion systems, Int. J.
Robust Nonlinear Control. 28 (2018), 1641–1652.[41] K. Wu, H. Sun, B. Yang and C.C. Lim, Finite-time boundary control for delay reaction-diffusion systems, Appl.
Math. Comput. 329 (2018), 52–63.
[42] K.N. Wu, Z. Wang, Y.Z. Wang and Z. Cui, Finite-time spatial sampled-data control for reaction-diffusion systems,
Circuits, Syst. Signal Process. 40 (2021), 4833–4849.
[43] Q. Zhu, Stabilization of stochastic nonlinear delay systems with exogenous disturbances and the event-triggered
feedback control, IEEE Trans. Autom. Control 64 (2019), 3764–3771.
Volume 13, Issue 2
July 2022
Pages 1821-1832
  • Receive Date: 27 January 2022
  • Revise Date: 23 February 2022
  • Accept Date: 09 May 2022