### Stability of a Variable Coefficient Star-Shaped Network with Distributed Delay

ZHANG Hai-E1,2, XU Gen-Qi1, CHEN Hao3, LI Min1

1. 1. School of Mathematics, Tianjin University, Tianjin 300350, China;
2. Department of Basic Science, Tangshan University, Tangshan 063000, China;
3. School of Mechatronical Engineering, Beijing Institute of Technology, Beijing 100081, China
• Received:2021-08-11 Revised:2022-05-14 Online:2022-11-25 Published:2022-12-23
• Supported by:
This research was supported by the National Natural Science Foundation of China under Grant Nos. 61773277 and 62073236.

ZHANG Hai-E, XU Gen-Qi, CHEN Hao, LI Min. Stability of a Variable Coefficient Star-Shaped Network with Distributed Delay[J]. Journal of Systems Science and Complexity, 2022, 35(6): 2077-2106.

The paper deals with the exponential stability problem of a variable coefficient star-shaped network, whose strings are coupled at a common end in a star-shaped configuration and the common connection of all strings can be moved. Two kinds of media materials with a component of viscous and another simply elastic are distributed on each string. Under suitable hypothesis on the coefficient functions $\mu_j(x)$ of damping terms and the kernels $\eta_j(s)$ of distributed delay terms, the well-posedness of the system is obtained by means of resolvent family theory. In addition, the allocation proportion of the two parts and the property of the material character functions are discussed when the star-shaped network is exponentially stable. Meanwhile, the sufficient condition of exponential stability is established. Numerical simulations are also included to verify the main results.
 [1] Dáger R and Zuazua E, Controllability of star-shaped networks of strings, Comptes Rendus de l’Académie des Sciences-Series I-Mathematics, 2001, 332(7): 621–626.[2] Dáger R and Zuazua E, Spectral boundary controllability of networks of strings, Comptes Rendus Mathematique, 2002, 334(7): 545–550.[3] Dáger R, Observation and control of vibrations in tree-shaped networks of strings, SIAM J. Control Optim., 2004, 43: 590–623.[4] Mercier D and Régnier V, Control of a network of Euler-Bernoulli beams, Journal of Mathematical Analysis and Applications, 2008, 342(2): 874–894.[5] Wang H Q, Bai W, and Zhao X D, Finite-time-prescribed performance-based adaptive fuzzy control for strict-feedback nonlinear systems with dynamic uncertainty and actuator faults, IEEE Transactions on Cybernetics, 2020, DOI: 10.1109/TCYB.2020.3046316.[6] Wang H Q, Xu K, and Qiu J B, Event-triggered adaptive fuzzy fixed-time tracking control for a class of nonstrict-feedback nonlinear systems, IEEE Transactions on Circuits and Systems I: Regular Papers, 2021, 68(7): 3058–3068.[7] Li Y, Yang T, and Tong S, Adaptive neural networks finite-time optimal control for a class of nonlinear systems, IEEE Transactions on Neural Networks and Learning Systems, 2020, 31(11): 4451–4460.[8] Nicaise S and Valein J, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks, Networks and Heterogeneous Media, 2007, 2: 425–479.[9] Xu G Q, Liu D Y, and Liu Y Q, Abstract second order hyperbolic system and applications to controlled network of strings, SIAM J. Conrtrol Opthim., 2008, 47(4): 1762–1784.[10] Xu G Q and Yung S P, Stability and riesz basis property of a star-shaped network of EulerBernoulli beams with joint damping, Networks and Heterogeneous Media, 2017, 3(4): 723–747.[11] Han Z J and Xu G Q, Stabilization and Riesz basis of a star-shaped network of Timoshenko beams, Journal of Dynamical and Control Systems, 2010, 16(2): 227–258.[12] Guo Y N and Xu G Q, Asymptotic stability and Riesz basis property for tree-shaped network of strings, Journal of Systems Science and Complexity, 2011, 24(2): 225–252.[13] Ammari K and Jellouli M, Stabilization of star-shaped networks of strings, Differential Integral Equations, 2004, 17(11–12): 1395–1410.[14] Ammari K, Jellouli M, and Khenissi M, Stabilization of generic trees of strings, Journal of Dynamical and Control Systems, 2005, 11(2): 177–193.[15] Ammari K and Crepeau E, Well-posedness and stabilization of the Benjamin-Bona-Mahony equation on star-shaped networks, Systems & Control Letters, 2019, 127: 39–43.[16] Li H T and Wang Y Z, Lyapunov-based stability and construction of lyapunov functions for Boolean networks, SIAM J. Control Optim., 2017, 55(6): 3437–3457.[17] Chen G, Fulling S A, Narcowich F J, et al., Exponential decay of energy of evolution equations with locally distributed damping, SIAM J. Appl. Math., 1991, 2017, 51(1): 266–301.[18] AliMehmeti F and Regnier R, Splitting of energy of dispersive wave in a start-shaped network, Z. Angew. Math. Mech., 2003, 83(2): 105–118.[19] Mironchenkoy A and Prieur C, Input-to-state stability of infinite-dimensional systems: Recent results and open questions, SIAM Rev., 2020, 62(3): 529–614.[20] AliMehmeti F and Regnier R, Splitting of energy of dispersive wave in a star-shaped network, Z. Angew. Math. Mech., 2003, 83(2): 105–118.[21] Han Z J and Zuazua E, Decay rates for elastic-thermoelastic star-shaped networks, Networks and Heterogeneous Media, 2012, 12(3): 461–488.[22] Munoz Rivera J E and Portillo Oquendo H, The transmission problem for thermoelastic beams, Journal of Thermal Stresses, 2001, 24: 1137–1158.[23] Marzocchi A, Munoz Rivera J E, and Naso M G, Asymptotic behaviour and expinential stability for a transmission problem in thermoelasticity, Math. Meth. Appl. Sci., 2002, 25: 955–980.[24] Messaoudi S A and Said-Houari B, Energy decay in a transmission problem in thermoelasticity of type III, IMA J. Appl. Math., 2009, 74: 344–360.[25] Malacarne A and Rivera J, Lack of exponential stability to Timoshenko system with viscoelastic Kelvin-Voigt type, Z. Angew. Math. Mech., 2016, 67: 67.[26] Kirane M and Said-Houari B, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 2011, 62: 1065–1082.[27] Liu W J, General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback, Journal of Mathematical Physics, 2013, 54(4): 1–11.[28] Vansickle J, Attrition in distributed delay models, IEEE Transactions on Systems Man & Cybernetics, 1977, 7(9): 635–638.[29] Zheng F and Frank P M, Robust control of uncertain distributed delay systems with application to the stabilization of combustion in rocket motor chambers, Automatica, 2002, 38: 487–497.[30] Liu G W, Well-posedness and exponential decay of solutions for a transmission problem with distributed delay, Electronic Journal of Differential Equations, 2017, 2017(174): 1–13.[31] Nicaise S and Pignotti C, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 2008, 21(9–10): 935–958.[32] Saravanakumar R, Rajchakit G, Ahn C K, et al., Exponential stability, passivity, and dissipativity analysis of generalized neural networks with mixed time-varying delays, IEEE Transactions on Systems Man & Cybernetics Systems, 2019, 49(2): 395–405.[33] Iswarya M, Raja R, Rajchakit G, et al., Existence, uniqueness and exponential stability of periodic solution for discrete-time delayed bam neural networks based on coincidence degree theory and graph theoretic method, Mathematics, 2019, 7(11): 1055.[34] Humphries U, Rajchakit G, Kaewmesri P, et al., Stochastic memristive quaternion-valued neural networks with time delays: An analysis on mean square exponential input-to-state stability, Mathematics, 2020, 8(5): 815.[35] Rajchakita G, Sriramanb R, Limc C, et al., Existence, uniqueness and global stability of Cliffordvalued neutral-type neural networks with time delays, Mathematics and Computers in Simulation, 2021, DOI: 10.1016/j.matcom.2021.02.023.[36] Rajchakit G, Sriraman R, Boonsatit N, et al., Global exponential stability of Clifford-valued neural networks with time-varying delays and impulsive effects, Advances in Difference Equations, 2021, 2021: 208.[37] Rajchakit G, Sriraman R, Boonsatit N, et al., Exponential stability in the Lagrange sense for Clifford-valued recurrent neural networks with time delays, Advances in Difference Equations, 2021, 2021: 256.[38] Xu G Q, Resolvent family for evolution process with memory, Mathematische Nachrichten, 2022, DOI: 10.1002/mana.202100203.
 [1] YANG Kunyi,REN Xiang,ZHANG Jie. Output Feedback Stabilization of an Unstable Wave Equation with Observations Subject to Time Delay [J]. Journal of Systems Science and Complexity, 2016, 29(1): 99-118. [2] ZHANG Xin. Reliability Analysis of a Cold Standby Repairable System with Repairman Extra Work [J]. Journal of Systems Science and Complexity, 2015, 28(5): 1015-1032. [3] ZHAO Dongxia , WANG Junmin. SPECTRAL ANALYSIS AND STABILIZATION OF A COUPLED WAVE-ODE SYSTEM [J]. Journal of Systems Science and Complexity, 2014, 27(3): 463-475. [4] BAO Leping , FEI Shumin, YU Lei. EXPONENTIAL STABILITY OF LINEAR DISTRIBUTED PARAMETER SWITCHED SYSTEMS WITH TIME-DELAY [J]. Journal of Systems Science and Complexity, 2014, 27(2): 263-275. [5] Yuanling NIU, Chengjian ZHANG. ALMOST SURE AND MOMENT EXPONENTIAL STABILITY OF PREDICTOR-CORRECTOR METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS [J]. Journal of Systems Science and Complexity, 2012, 25(4): 736-743. [6] Lina GUO , Houbao XU , Chao GAO , Guangtian ZHU. FURTHER RESEARCH OF A NEW KIND OF SERIES REPAIRABLE SYSTEM [J]. Journal of Systems Science and Complexity, 2012, 25(4): 744-758. [7] Zhaoqiang GE;Guangtian ZHU;Dexing FENG. DEGENERATE SEMI-GROUP METHODS FOR THE EXPONENTIAL STABILITY OF THE FIRST ORDER SINGULAR DISTRIBUTED PARAMETER SYSTEMS [J]. Journal of Systems Science and Complexity, 2008, 21(2): 260-266. [8] Weiwei HU;Houbao XU;Jingyuan YU;Guangtian ZHU. EXPONENTIAL STABILITY OF A REPARABLE MULTI-STATE DEVICE [J]. Journal of Systems Science and Complexity, 2007, 20(3): 437-443. [9] YAN Qingxu;HOU S. H.;HUANG G.D.;WAN Li. STABILIZATION OF NONUNIFORM TIMOSHENKO BEAM WITH COUPLED LOCALLY DISTRIBUTED FEEDBACKS [J]. Journal of Systems Science and Complexity, 2005, 18(3): 419-428. [10] Zhen Guo CHEN;Qing Xu YAN;Zhao Qi LI. BOUNDARY STABILIZATION OF NONUNIFORM TIMOSHENKO BEAM WITH ROTOR INERTIA AT THE TIP [J]. Journal of Systems Science and Complexity, 2004, 14(2): 176-187.
Viewed
Full text

Abstract