高维变区域上抛物方程的镇定

doi:10.12341/jssms22371

系统科学与数学 ›› 2022, Vol. 42 ›› Issue (11): 2914-2927.DOI: 10.12341/jssms22371

• • 上一篇

高维变区域上抛物方程的镇定

郑国杰¹, 孙怡悦², 仝荣花³

1. 宁波财经学院数字技术与工程学院, 宁波 315175;
2. 河南师范大学数学与信息科学学院, 新乡 453007;
3. 确山县第一高级中学, 驻马店 463299

收稿日期:2022-05-20 修回日期:2022-06-21 发布日期:2022-12-13
通讯作者: 郑国杰, Email: guojiezheng@whu.edu.cn
基金资助:
河南省自然科学基金项目(202300410248),浙江省自然科学基金重点项目(LZ21A010001)资助课题.

PDF ( 5 ) 引用

郑国杰, 孙怡悦, 仝荣花. 高维变区域上抛物方程的镇定[J]. 系统科学与数学, 2022, 42(11): 2914-2927.

ZHENG Guojie, SUN Yiyue, TONG Ronghua. Stabilization of a Parabolic Equation on Multi-Dimensional Time-Varying Domains[J]. Journal of Systems Science and Mathematical Sciences, 2022, 42(11): 2914-2927.

Stabilization of a Parabolic Equation on Multi-Dimensional Time-Varying Domains

ZHENG Guojie¹, SUN Yiyue², TONG Ronghua³

1. College of Digital Technology and Engineering, Ningbo University of Finance & Economics, Ningbo 315175;
2. College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007;
3. Queshan No. 1 Middle School, Zhumadian 463299

Received:2022-05-20 Revised:2022-06-21 Published:2022-12-13

文章研究了高维变区域上不稳定的抛物方程的镇定问题.在研究过程中首先讨论了变区域上抛物方程的适定性.其次运用了谱分析的方法,研究了变区域上拉普拉斯算子的第一特征值的单调性, 连续性等性质;再次借助于解的正则性, 证明了一个重要的能量等式;接着运用内部控制的技巧, 有效镇定了变区域上不稳定的抛物方程;最后给出一个例子和相应的数值实验来验证理论的有效性.

关键词： 抛物方程, 时变区域, 指数稳定, 内部控制

This paper aims to stabilize an unstable parabolic equation on multi-dimensional variable domains. We first investigate the well-posedness of the parabolic equation on the time-varying domains. Secondly, by using the method of spectral analysis, we study the monotonicity and continuity of the first eigenvalue of the Laplacian operator on the time-varying domains; then we obtain an important energy equation by the regularity of the solution. Next, we can stabilize the unstable parabolic equation with the help of internal control. Finally, an example and its numerical results are given to verify the validity of the proposed theory.

Key words: Parabolic equation, time-varying domain, exponential stability, internal control

MR(2010)主题分类:

35K15
34H15

分享此文:

[1] 郭宝珠, 柴树根. 无穷维线性系统控制理论. 北京: 科学出版社, 2012. (Guo B Z, Chai S G. Control Theory of Linear Infinite-Dimensional Systems. Beijing: Science Press, 2012.)
[2] Smyshlyaev A, Krstic M. On control design for PDEs with space-dependent diffusivity or time-dependent reactivity. Automatica, 2005, 41 (9): 1601-1608.
[3] Liu W. Boundary feedback stabilization of an unstable heat equation. SIAM Journal on Control and Optimization, 2003, 42 (3): 1033-1043.
[4]Jia J. Boundary feedback control of unstable heat equation with space and time dependent coefficient. Mathematics, 2006.
[5]Cannon J R. The One-Dimensional Heat Equation. Cambridge: Cambridge University Press, 1984.
[6]Zheng G, Li J. Stabilization for the multi-dimensional heat equation with disturbance on the controller. Automatica, 2017, 82 : 319-323.
[7]Vazqueza R, Krstic M. Control of 1-D parabolic PDEs with Volterra nonlinearities, Part I: Design. Automatica, 2008, 44 (12): 2778-2790.
[8]Smyshlyaev A, Krstic M. Backstepping observers for a class of parabolic PDEs. Systems & Control Letters, 2005, 54 (7): 613-625.
[9] 汤玉晶. 非柱状区域上一维热方程的稳定性问题. 硕士论文. 东北师范大学, 吉林, 2016. (Tang Y J. Stability problem of a one-dimensional heat equation in non-cylindrical domains. Master Thesis. Northeast Normal University, Jilin, 2016.)
[10] Antonopoulou D C, Karali G D, Yip N K. On the parabolic Stefan problem for Ostwald ripening with kinetic undercooling and inhomogeneous driving force. Journal of Differential Equations, 2012, 252 (9): 4679-4718.
[11] Isi M, Farr W M, Giesler M, et al. Testing the black-hole area law with GW150914. Physical Review Letters, 2021, 127 (1): 011103.
[12] Wang P K C. Stabilization and control of distributed systems with time-dependent spatial domains. Journal of Optimization Theory and Applications, 1990, 65 (2): 331-362.
[13] Izadi M, Abdollahi J, Dubljevic S S. PDE backstepping control of one-dimensional heat equation with time-varying domain. Automatica, 2015, 54 : 41-48.
[14]Menezes S B D, Limaco J, Medeiros L A. Remarks on null controllability for semilinear heat equation in moving domains. Electronic Journal of Qualitative Theory of Differential Equations, 2003, 16 (16): 1-32.
[15] Cui L, Liu X, Gao H. Exact controllability for a one-dimensional wave equation in non-cylindrical domains. Journal of Mathematical Analysis and Applications, 2013, 402 (2): 612-625.
[16]Límaco J, Medeiros L A, Zuazua E. Existence, uniqueness and controllability for parabolic equations in noncylindrical domains. Mat Contemp, 2002, 23 : 49-70.
[17] Kloeden P E, Real J, Sun C. Pullback attractors for a semilinear heat equation on time-varying domains. Journal of Differential Equations, 2009, 246 (12): 4702-4730.
[18]Evans L C. Partial Differential Equations. American Mathematical Society, 1964.
[19]Barbu V, Lefter C, Tessitore G. A note on the stabilizability of stochastic heat equations with multiplicative noise. Comptes Rendus Mathematique, 2002, 334 (4): 311-316.
[20] Yosida K. Functional Analysis. Germany: Springer-Verlag, 1978.

[1]	祁雪萍, 阿不都克热木·阿吉. 具有小修和一般型更换策略的多状态退化系统的定性分析[J]. 系统科学与数学, 2021, 41(9): 2571-2594.
[2]	张小英, 王平, 冯红银萍. 常微分方程-薛定谔方程耦合系统的输出反馈镇定[J]. 系统科学与数学, 2021, 41(4): 887-897.
[3]	支霞, 冯红银萍. 输入带有时滞的线性系统的镇定[J]. 系统科学与数学, 2021, 41(1): 17-23.
[4]	刘建康，李欢欢. Robin型边界阻尼波动方程的一致指数稳定逼近[J]. 系统科学与数学, 2020, 40(4): 599-611.
[5]	赵云波，姚俊毅，倪洪杰. 多径路由网络化控制系统的路径调度与控制器协同设计[J]. 系统科学与数学, 2019, 39(4): 507-521.
[6]	李钊，李树勇. 比较原理与一类具有时滞和马尔科夫切换的随机抛物方程组的稳定性分析[J]. 系统科学与数学, 2019, 39(4): 648-658.
[7]	彭树霞，许跟起. 带有时滞控制的一维热传导方程的参数化控制器设计与稳定性研究[J]. 系统科学与数学, 2019, 39(1): 1-14.
[8]	徐姝婕，程培，贲顺琦. 不确定脉冲随机系统的几乎必然指数稳定和随机镇定[J]. 系统科学与数学, 2017, 37(1): 10-22.
[9]	邵汉永，张丹，赵建荣. 鲁棒采样依赖指数稳定性分析新方法[J]. 系统科学与数学, 2016, 36(7): 986-994.
[10]	李艳，胡军浩，沈轶. 反应扩散递归神经网络全局指数稳定的鲁棒分析[J]. 系统科学与数学, 2015, 35(2): 142-157.
[11]	董亚丽，左世凯，刘婉军. 带有时变时滞的中立型神经网络的鲁棒指数稳定性[J]. 系统科学与数学, 2014, 34(11): 1391-1400.
[12]	章春国，谷尚武，姜敬华. 具有Boltzmann阻尼的系统的稳定性[J]. 系统科学与数学, 2013, 33(7): 807-817.
[13]	吴艳蕾. 中立型带跳的双干\随机微分方程解的$\bm L^{\bm p}$估计[J]. 系统科学与数学, 2013, 33(4): 496-505.
[14]	李亚军，邓飞其. 中立型带跳不确定变时滞随机系统鲁棒指数稳定[J]. 系统科学与数学, 2012, 32(7): 821-830.
[15]	袁燕飞;葛照强. 一阶广义分布参数系统的指数稳定性[J]. 系统科学与数学, 2010, 30(3): 315-322.

阅读次数

全文

摘要