基于边界采样控制的随机反应扩散系统稳定性

王云竹

doi:10.21656/1000-0887.450309

基于边界采样控制的随机反应扩散系统稳定性

doi: 10.21656/1000-0887.450309

王云竹

浙大城市学院理学基础教育中心，杭州 310015

详细信息

作者简介:
王云竹(1992—)，女，博士(E-mail: hzcu_wyz@163.com)

中图分类号: O231.3
计量
- 文章访问数: 151
- HTML全文浏览量: 58
- PDF下载量: 29
- 被引次数: 0
出版历程
- 收稿日期: 2024-11-18
- 修回日期: 2025-12-15
- 刊出日期: 2026-04-01

Stability of Stochastic Reaction-Diffusion Systems via Boundary Sampled-Data Control

WANG Yunzhu

Foundation Science Education Center, Hangzhou City University, Hangzhou 310015, P.R.China

摘要

摘要: 利用边界采样控制讨论了随机反应扩散系统(stochastic reaction-diffusion system，SRDS)稳定性问题.当系统状态可以全部获取时，设计了一个边界采样控制器(boundary sampling controller，BSC)，构建了与采样间隔相关的分段不连续Lyapunov函数.对于SRDS，利用空间积分型Wirtinger不等式和同构离散变换，得到了矩阵不等式形式的均方指数稳定和鲁棒均方指数稳定的充分条件.当系统状态无法完全获得时，提出了一种基于观测器的边界采样控制策略，分别得到了系统均方指数稳定和鲁棒均方指数稳定的研究结果.最后，通过三个数值例子验证了所提方法的可行性.
- 边界采样控制 /
- 稳定性 /
- 随机反应扩散系统 /
- 观测器
Abstract: The stability of stochastic reaction-diffusion systems under boundary sampling control was discussed. With the fully accessible system state, a boundary sampled-data controller was proposed, and a piecewise discontinuous Lyapunov function related to the sampling interval was constructed. For stochastic reaction-diffusion systems, sufficient conditions for mean-square exponential stability, both in nominal and robust settings, were obtained with the Wirtinger inequality for spatial integrals and isomorphic discrete transformations, in the form of matrix inequalities. With the not fully accessible system state, an observer-based boundary sampled-data control strategy was proposed, and results for the mean-square exponential stability and the robust mean-square exponential stability of the system were obtained, respectively. Finally, the feasibility of the proposed methods was demonstrated through 3 numerical examples.
- boundary sampled-data control /
- stability /
- stochastic reaction-diffusion system /
- observer

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图 1 系统(54)的状态响应(有控制)

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Figure 1. The state responses of system (54) (with control)

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图 2 系统(54)的状态响应(无控制)

Figure 2. The state responses of system (54) (without control)

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图 3 系统(55)的状态响应(有控制)

Figure 3. The state responses of system (55) (with control)

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图 4 系统(55)的状态响应(无控制)

Figure 4. The state responses of system (55) (without control)

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图 5 系统(56)的状态响应(有控制)

Figure 5. The state responses of system (56) (with control)

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图 6 系统(56)的状态响应(无控制)

Figure 6. The state responses of system (56) (without control)

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参考文献(33)

[1]	WU K N, LIU X Z, SHI P, et al. Boundary control of linear stochastic reaction-diffusion systems[J]. International Journal of Robust and Nonlinear Control, 2019, 29 (1): 268-282. doi: 10.1002/rnc.4386
[2]	董浩祖, 肖敏, 丁洁, 等. 具有饱和控制的时滞反应扩散谣言传播模型[J]. 系统科学与数学, 2024, 44 (1): 1-16. DONG Haozu, XIAO Min, DING Jie, et al. Time-delay reaction-diffusion rumor propagation model with saturation control[J]. Journal of Systems Science and Mathematical Sciences, 2024, 44 (1): 1-16. (in Chinese)
[3]	杜彩虹. 具有蚊子叮咬偏好性的扩散疟疾模型的动力学[J]. 应用数学和力学, 2023, 44 (3): 345-354. doi: 10.21656/1000-0887.430095 DU Caihong. Dynamics of a diffusion malaria model with vector-bias[J]. Applied Mathematics and Mechanics, 2023, 44 (3): 345-354. (in Chinese) doi: 10.21656/1000-0887.430095
[4]	徐鹤丽, 薛文宁, 高建国. 具有反应扩散的随机肿瘤免疫模型接近最优控制研究[J]. 应用数学, 2024, 37 (2): 439-455. XU Heli, XUE Wenning, GAO Jianguo. Near-optimal control of stochastic tumor immune model with reaction diffusion[J]. Mathematica Applicata, 2024, 37 (2): 439-455. (in Chinese)
[5]	PAN L J, CAO J D, ALSAEDI A. Stability of reaction-diffusion systems with stochastic switching[J]. Nonlinear Analysis: Modelling and Control, 2019, 24 (3): 315-331. doi: 10.15388/NA.2019.3.1
[6]	WU K N, WANG J. Mixed H₂/H_∞ control of synchronization for coupled partial differential systems[J]. International Journal of Robust and Nonlinear Control, 2017, 27 (9): 1397-1418. doi: 10.1002/rnc.3722
[7]	VAZQUEZ R, KRSTIC M. Boundary control of coupled reaction-advection-diffusion systems with spatially-varying coefficients[J]. IEEE Transactions on Automatic Control, 2017, 62 (4): 2026-2033. doi: 10.1109/TAC.2016.2590506
[8]	WU K N, WANG J, LIM C C. Synchronization of stochastic reaction-diffusion systems via boundary control[J]. Nonlinear Dynamics, 2018, 94 (3): 1763-1773. doi: 10.1007/s11071-018-4455-z
[9]	LEE J, MOON J H, LEE H J. Continuous-time synthesizing robust sampled-data dynamic output-feedback controllers for uncertain nonlinear systems in Takagi-Sugeno form: a descriptor representation approach[J]. Information Sciences, 2021, 565 : 456-468. doi: 10.1016/j.ins.2021.02.032
[10]	ZHU B, SUO M L, CHEN Y, et al. Mixed H_∞ and passivity control for a class of stochastic nonlinear sampled-data systems[J]. Journal of the Franklin Institute, 2018, 355 (7): 3310-3329. doi: 10.1016/j.jfranklin.2018.01.040
[11]	ZHAO F L, WANG Z P, WU H N, et al. Robust event-triggered sampled-data fuzzy control with non-fragile for nonlinear delayed distributed parameter systems[J]. International Journal of Fuzzy Systems, 2023, 25 (6): 2312-2325. doi: 10.1007/s40815-023-01505-1
[12]	WANG Z P, CHEN B M, QIAO J F, et al. Fuzzy boundary sampled-data control for nonlinear DPSs with random time-varying delays[J]. IEEE Transactions on Fuzzy Systems, 2024, 32 (10): 5872-5885. doi: 10.1109/TFUZZ.2024.3432795
[13]	WANG Z P, LI Q Q, QIAO J F, et al. Fuzzy boundary sampled-data control for nonlinear parabolic DPSs[J]. IEEE Transactions on Cybernetics, 2024, 54 (6): 3565-3576. doi: 10.1109/TCYB.2023.3312135
[14]	贺彦君, 马伟伟, 池小波. 网络化非周期采样控制系统的主动时间滞后控制: 随机脉冲切换系统方法[J]. 应用数学和力学, 2021, 42 (4): 422-430. doi: 10.21656/1000-0887.410213 HE Yanjun, MA Weiwei, CHI Xiaobo. Active time-delay control of networked control systems with aperiodic sampling: a stochastic impulsive switched system approach[J]. Applied Mathematics and Mechanics, 2021, 42 (4): 422-430. (in Chinese) doi: 10.21656/1000-0887.410213
[15]	KARAFYLLIS I, KRSTIC M. Sampled-data boundary feedback control of 1-D parabolic PDEs[J]. Automatica, 2018, 87 : 226-237. doi: 10.1016/j.automatica.2017.10.006
[16]	LIU H B, HU P. Boundary sampled-data feedback stabilization for parabolic equations[J]. Systems & Control Letters, 2020, 136 : 104618.
[17]	AHMED-ALI T, GIRI F, KARAFYLLIS I, et al. Sampled boundary observer for strict-feedback nonlinear ODE systems with parabolic PDE sensor[J]. Automatica, 2019, 101 : 439-449. doi: 10.1016/j.automatica.2018.12.014
[18]	FERRANTE F, CRISTOFARO A, PRIEUR C. Boundary observer design for cascaded ODE: hyperbolic PDE systems: a matrix inequalities approach[J]. Automatica, 2020, 119 : 109027. doi: 10.1016/j.automatica.2020.109027
[19]	XIE L, KHARGONEKAR P P. Lyapunov-based adaptive state estimation for a class of nonlinear stochastic systems[J]. Automatica, 2012, 48 (7): 1423-1431. doi: 10.1016/j.automatica.2012.05.002
[20]	ZHAO Y, QIU J B, XU S Y, et al. Boundary observer-based control for hyperbolic PDE-ODE cascade systems with stochastic jumps[J]. Automatica, 2020, 119 : 109089. doi: 10.1016/j.automatica.2020.109089
[21]	YIN Y Y, SHI P, LIU F, et al. Observer-based H_∞ control on stochastic nonlinear systems with time-delay and actuator nonlinearity[J]. Journal of the Franklin Institute, 2013, 350 (6): 1388-1405. doi: 10.1016/j.jfranklin.2013.02.007
[22]	SHAO H Y, HAN Q L, ZHANG Z Q, et al. Sampling-interval-dependent stability for sampled-data systems with state quantization[J]. International Journal of Robust and Nonlinear Control, 2014, 24 (17): 2995-3008. doi: 10.1002/rnc.3038
[23]	XIAO S P, LIAN H H, TEO K L, et al. A new Lyapunov functional approach to sampled-data synchronization control for delayed neural networks[J]. Journal of the Franklin Institute, 2018, 355 (17): 8857-8873. doi: 10.1016/j.jfranklin.2018.09.022
[24]	LEE T H, PARK J H. Stability analysis of sampled-data systems via free-matrix-based time-dependent discontinuous Lyapunov approach[J]. IEEE Transactions on Automatic Control, 2017, 62 (7): 3653-3657. doi: 10.1109/TAC.2017.2670786
[25]	FRIDMAN E, BLIGHOVSKY A. Robust sampled-data control of a class of semilinear parabolic systems[J]. Automatica, 2012, 48 (5): 826-836. doi: 10.1016/j.automatica.2012.02.006
[26]	LUO S X, DENG F Q. A note on delay-dependent stability of Itô-type stochastic time-delay systems[J]. Automatica, 2019, 105 : 443-447. doi: 10.1016/j.automatica.2019.03.005
[27]	CHEN W H, LUO S X, ZHENG W X. Sampled-data distributed H_∞ control of a class of 1-D parabolic systems under spatially point measurements[J]. Journal of the Franklin Institute, 2017, 354 (1): 197-214. doi: 10.1016/j.jfranklin.2016.09.028
[28]	ZHANG X M, WU H N. H_∞ boundary control for a class of nonlinear stochastic parabolic distributed parameter systems[J]. International Journal of Robust and Nonlinear Control, 2019, 29 (14): 4665-4680. doi: 10.1002/rnc.4646
[29]	WU Z G, PARK J H, SU H Y, et al. Discontinuous Lyapunov functional approach to synchronization of time-delay neural networks using sampled-data[J]. Nonlinear Dynamics, 2012, 69 (4): 2021-2030. doi: 10.1007/s11071-012-0404-4
[30]	XING M L, DENG F Q. Dynamic output feedback control for stochastic networked control systems based on a periodic event-triggered mechanism[J]. Transactions of the Institute of Measurement and Control, 2019, 41 (7): 1975-1984. doi: 10.1177/0142331218792415
[31]	LIU X Z, WU K N, ZHANG W H. Mean square finite-time boundary stabilisation and H_∞ boundary control for stochastic reaction-diffusion systems[J]. International Journal of Systems Science, 2019, 50 (7): 1388-1398. doi: 10.1080/00207721.2019.1615574
[32]	HAN X X, WU K N, DING X H, et al. Boundary control of stochastic reaction-diffusion systems with Markovian switching[J]. International Journal of Robust and Nonlinear Control, 2020, 30 (10): 4129-4148. doi: 10.1002/rnc.4992
[33]	WEI T D, WANG L S, WANG Y F. Existence, uniqueness and stability of mild solutions to stochastic reaction-diffusion Cohen-Grossberg neural networks with delays and Wiener processes[J]. Neurocomputing, 2017, 239 : 19-27. doi: 10.1016/j.neucom.2017.01.069