Controller design for chain-type wave network models with boundary delays

Yaru Xie; Ruiqing Gao; Yaru Xie; Ruiqing Gao

doi:10.3934/math.2025161

AIMS Mathematics

2025, Volume 10, Issue 2: 3484-3499. doi: 10.3934/math.2025161

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Research article

Controller design for chain-type wave network models with boundary delays

Yaru Xie ,
Ruiqing Gao ^,

Department of Mathematics, Civil Aviation University of China, Tianjin 300300, China

Received: 31 December 2024 Revised: 01 February 2025 Accepted: 17 February 2025 Published: 24 February 2025
MSC : 31B10, 35R02, 93C35, 93C43, 93D15

In this paper, we consider the design method of full-state integral feedback controllers, selecting appropriate target systems, establishing equivalent transformations, and completing the stability analysis of the original system while avoiding a large amount of complex calculations and proofs. This research attempts to establish a universally applicable controller design strategy that can not only eliminate the negative effects of system delays, but also ensure the closed-loop stability. Therefore, this paper further explores the applicability of this controller design strategy in complex PDE-coupled systems, seeks general patterns, and has further theoretical and practical value for the stability study of infinite-dimensional time-delay systems.
- chain wave network,
- input delay,
- controller design,
- exponential stabilization,
- PDE-coupled system
Citation: Yaru Xie, Ruiqing Gao. Controller design for chain-type wave network models with boundary delays[J]. AIMS Mathematics, 2025, 10(2): 3484-3499. doi: 10.3934/math.2025161

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Abstract

In this paper, we consider the design method of full-state integral feedback controllers, selecting appropriate target systems, establishing equivalent transformations, and completing the stability analysis of the original system while avoiding a large amount of complex calculations and proofs. This research attempts to establish a universally applicable controller design strategy that can not only eliminate the negative effects of system delays, but also ensure the closed-loop stability. Therefore, this paper further explores the applicability of this controller design strategy in complex PDE-coupled systems, seeks general patterns, and has further theoretical and practical value for the stability study of infinite-dimensional time-delay systems.

References

[1]	S. Manavi, E. Fattahi, T. Becker, A trial solution for imposing boundary conditions of partial differential equations in physics-informed neural networks, Eng. Appl. Artif. Intell., 127 (2024), 107236. https://doi.org/10.1016/j.engappai.2023.107236 doi: 10.1016/j.engappai.2023.107236
[2]	Y. Wang, Z. Lin, Y. Liao, H. Liu, H. Xie, Solving high-dimensional partial differential equations using tensor neural network and a posteriori error estimators, J. Sci. Comput., 101 (2024), 67. https://doi.org/10.1007/s10915-024-02700-4 doi: 10.1007/s10915-024-02700-4
[3]	K. Ammari, F. Shel, Stability of a tree-shaped network of strings and beams, In: Stability of Elastic Multi-Link Structures, 2021, 57–88. https://doi.org/10.1007/978-3-030-86351-7_4
[4]	A. Öchsner, Euler-bernoulli beam theory, In: Classical Beam Theories of Structural Mechanics, 2021, 7–66. https://doi.org/10.1007/978-3-030-76035-9_2
[5]	Y. Xie, G. Xu, The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls, Networks Heterog. Media, 11 (2016), 527–543. https://doi.org/10.3934/nhm.2016008 doi: 10.3934/nhm.2016008
[6]	D. Liu, L. Zhang, Genqi. Xu, Z. Han, Stabilization of one-dimensional wave equations coupled with an ode system on general tree-shaped networks, IMA J. Math. Control Inform., 32 (2015), 557–589. https://doi.org/10.1093/imamci/dnu008 doi: 10.1093/imamci/dnu008
[7]	D. Liu, G. Xu, Riesz basis and stability analysis of the feedback controlled networks of 1-d wave equations, WSEAS Trans. Math., 11 (2012), 418–439.
[8]	Y. Zhang, Z. Han, G. Xu, Stability and spectral properties of general tree-shaped wave networks with variable coefficients, Acta Appl. Math., 164 (2019), 219–249. https://doi.org/10.1007/s10440-018-00236-y doi: 10.1007/s10440-018-00236-y
[9]	P. Yao, Energy decay for the cauchy problem of the linear wave equation of variable coefficients with dissipation, Chin. Ann. Math. Ser. B, 31 (2010), 59–70. https://doi.org/10.1007/s11401-008-0421-2 doi: 10.1007/s11401-008-0421-2
[10]	G. Xu, S. Yung, L. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM: Control Optim Calculus Var., 12 (2006), 770–785. https://doi.org/10.1051/cocv:2006021 doi: 10.1051/cocv:2006021
[11]	Y. Shang, G. Xu, Stabilization of an euler-bernoulli beam with input delay in the boundary control, Syst. Control Lett., 61 (2012), 1069–1078. https://doi.org/10.1016/j.sysconle.2012.07.012 doi: 10.1016/j.sysconle.2012.07.012
[12]	Y. Shang, G. Xu, Dynamic feedback control and exponential stabilization of a compound system, J. Math. Anal. Appl., 422 (2015), 858–879. https://doi.org/10.1016/j.jmaa.2014.09.013 doi: 10.1016/j.jmaa.2014.09.013
[13]	H. Wang, G. Xu, Exponential stabilization of 1-d wave equation with input delay, WSEAS Trans. Math., 12 (2013), 1001–1013.
[14]	Y. Guo, Y. Chen, G. Xu, Y. Zhang, Exponential stabilization of variable coefficient wave equations in a generic tree with small time-delays in the nodal feedbacks, J. Math. Anal. Appl., 395 (2012), 727–746. https://doi.org/10.1016/j.jmaa.2012.05.079 doi: 10.1016/j.jmaa.2012.05.079
[15]	X. Liu, G. Xu, Exponential stabilization for timoshenko beam with different delays in the internal feedback, J. Syst. Sci. Math. Sci., 38 (2018), 131.
[16]	G. Xu, A. J. Rahmati, F. Badpar, Dynamic feedback stabilization of timoshenko beam with internal input delays, WSEAS Trans. Math., 17 (2018), 101–112.
[17]	Y. Shang, G. Xu, X. Li, Output-based stabilization for a one-dimensional wave equation with distributed input delay in the boundary control, IMA J. Math. Control Inform., 33 (2016), 95–119. https://doi.org/10.1093/imamci/dnu030 doi: 10.1093/imamci/dnu030
[18]	H. Cui, Z. Han, G. Xu, Stabilization for schrödinger equation with a time delay in the boundary input, Appl. Anal., 95 (2016), 963–977. https://doi.org/10.1080/00036811.2015.1047830 doi: 10.1080/00036811.2015.1047830
[19]	S. Cox, E. Zuazua, The rate at which energy decays in a damped string, Commun. PDE, 19 (1994), 213–243. https://doi.org/10.1080/03605309408821015 doi: 10.1080/03605309408821015
[20]	M. Shubov, Nonselfadjoint operators generated by the equation of a nonhomogeneous damped string, Trans. Amer. Math. Soc., 349 (1997), 4481–4499. https://doi.org/10.1090/S0002-9947-97-02044-8 doi: 10.1090/S0002-9947-97-02044-8
[21]	C. Prieur, E. Trélat, Feedback stabilization of a 1-d linear reaction–diffusion equation with delay boundary control, IEEE Trans. Autom. Control, 64 (2018), 1415–1425. https://doi.org/10.1109/TAC.2018.2849560 doi: 10.1109/TAC.2018.2849560
[22]	X. Feng, G. Xu, Y. Chen, Rapid stabilisation of an euler–bernoulli beam with the internal delay control, Int. J. Control, 92 (2019), 42–55. https://doi.org/10.1080/00207179.2017.1286693 doi: 10.1080/00207179.2017.1286693
[23]	R. Assel, M. Ghazel, Finite time stabilization of the waves on an infinite network, J. Math. Anal. Appl., 503 (2021), 125303. https://doi.org/10.1016/j.jmaa.2021.125303 doi: 10.1016/j.jmaa.2021.125303
[24]	K. E. Mufti, R. Yahia, Polynomial stability in viscoelastic network of strings, Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), 1421–1438. http://doi.org/10.3934/dcdss.2022073 doi: 10.3934/dcdss.2022073
[25]	A. Hayek, S. Nicaise, Z. Salloum, A. Wehbe, Exponential and polynomial stability results for networks of elastic and thermo-elastic rods, Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), 1183. https://doi.org/10.3934/dcdss.2021142 doi: 10.3934/dcdss.2021142
[26]	G. Xu, L. Zhang, Uniform stabilization of 1-d coupled wave equations with anti-dampings and joint delayed control, SIAM J. Control Optim., 58 (202), 3161–3184. https://doi.org/10.1137/19M1289145 doi: 10.1137/19M1289145
[27]	K. Ammari, B. Chentouf, N. Smaoui, Internal feedback stabilization of multi-dimensional wave equations with a boundary delay: a numerical study, Bound. Value Probl., 2022 (2022), 8. https://doi.org/10.1186/s13661-022-01589-y doi: 10.1186/s13661-022-01589-y
[28]	X. Wang, G. Xu, Uniform stabilization of a wave equation with partial dirichlet delayed control, Evol. Equ. Control Theory, 9 (2020), 509. https://doi.org/10.3934/eect.2020022 doi: 10.3934/eect.2020022
[29]	A. Pazy, Semigroups of linear operators and applications to partial differential equations, Berlin: Springer Science & Business Media, 2012.
[30]	Y. Xie, Z. Han, G. Xu, Stabilization of serially connected hybrid pde–ode system with unknown external disturbances, Appl. Anal., 98 (2019), 718–734. https://doi.org/10.1080/00036811.2017.1400535 doi: 10.1080/00036811.2017.1400535

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