| Citation: |
LI Lingfei, LIANG Xiaoyue, ZHANG Xiaoyi, WANG Yueshan. The Stabilization of A Class of Nonlinear Degenerate Parabolic Equation[J]. Journal of South China Normal University (Natural Science Edition), 2024, 56(2): 100-104. DOI: 10.6054/j.jscnun.2024027
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The exponential stabilization of a class of nonlinear degenerate parabolic equation is investigated in this paper. The stabilization of the homogenous system is obtained by the method of lifting the boundary to the equation. Based on the imbedding theorems for the weighted Sobolev space, the exponential stabilization of the nonlinear system is established via the fixed point theorem.
| [1] |
LIN P, GAO H, LIU X. Some results on a nonlinear degenerate parabolic system by bilinear control[J]. Journal of Mathematical Analysis Application, 2007, 326(2): 1149-1160. doi: 10.1016/j.jmaa.2006.03.079
|
| [2] |
解春雷, 杜润梅. 一类半线性退化抛物方程在边界控制函数作用下的近似能控性[J]. 吉林大学学报(理学版), 2021, 59(3): 563-567. https://www.cnki.com.cn/Article/CJFDTOTAL-JLDX202103020.htm
XIE C L, DU R M. Approximate controllability of a class of semilinear degenerate parabolic equations with boundary control functions[J]. Journal of Jilin University(Science Edition), 2021, 59(3): 563-567. https://www.cnki.com.cn/Article/CJFDTOTAL-JLDX202103020.htm
|
| [3] |
CANNARSA P, FRAGNELLI G, ROCCHETTI D. Controll-ability results for a class of one-dimensional degenerate parabolic problems in nondivergence form[J]. Journal of Evolution Equations, 2008, 8: 583-616. doi: 10.1007/s00028-008-0353-34
|
| [4] |
WANG C P. Approximate controllability of a class of degenerate systems[J]. Applied Mathematics Computation, 2008, 203(1): 447-456. doi: 10.1016/j.amc.2008.04.056
|
| [5] |
CANNARSA P, MARTINEZ P, VANCOSTENOBLE J. Persistent regional null controllability for a class of degene-rate parabolic equations[J]. Communications on Pure and Applied Analysis, 2004, 3(4): 607-635.
|
| [6] |
CANNARSA P, FRAGNELLI G, VANCOSTENOBLE J. Regional controllability of semilinear degenerate parabolic equations in bounded domains[J]. Journal of Mathematical Analysis and Applications, 2006, 320(2): 804-818. doi: 10.1016/j.jmaa.2005.07.006
|
| [7] |
GUEYE M. Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations[J]. SIAM Journal of Control and Optimization, 2014, 52(4): 2037-2054. doi: 10.1137/120901374
|
| [8] |
WANG C P. Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy[J]. Discrete and Continuous Dynamical Systems, 2016, 36(2): 1041-1060.
|
| [9] |
GAO H, LI L F, LIU Z Y. Stability of degenerate heat equation in non-cylindrical/cylindrical domain[J]. Zeitschrift für Angewandte Mathematik und Physik, 2019, 70(4): 120/1-17.
|
| [10] |
BARBU V. Boundary stabilization of equilibrium solutions to parabolic equations[J]. IEEE Transactions on Automatic Control, 2013, 58(9): 2416-2420. doi: 10.1109/TAC.2013.2254013
|
| [11] |
CORON J M, NGUYEN H M. Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[J]. Archive for Rational Mechanics and Analysis, 2017, 225(3): 993-1023. doi: 10.1007/s00205-017-1119-y
|
| [12] |
CORON J M, LV Q. Fredholm transform and local rapid stabilization for a Kuramoto-Sivashinsky equation[J]. Journal of Differential Equations, 2015, 259(8): 3683-3729. doi: 10.1016/j.jde.2015.05.001
|
| [13] |
KRSTIC M, GUO B Z, BALOGH A, et al. Output-feedback stabilization of an unstable wave equation[J]. Automatica, 2008, 44(1): 63-74. doi: 10.1016/j.automatica.2007.05.012
|
| [14] |
GAGNON L, HAYAT A, XIANG S Q, et al. Fredholm transformation on Laplacian and rapid stabilization for the heat equation[J]. Journal of Functional Analysis, 2022, 283(12): 109664/1-57.
|
| [15] |
CORON J M, XIANG S Q. Small-time global stabilization of the viscous Burgers equation with three scalar controls[J]. Journal de Mathématiques Pures et Appliquées, 2021, 151: 212-256. doi: 10.1016/j.matpur.2021.03.001
|
| [16] |
BARBU V, WANG G S. Feedback stabilization of semili-near heat equations[J]. Abstract and Applied Analysis, 2003, 12: 697-714.
|
| [17] |
TRIGGIANI R. Boundary feedback stabilizability of parabolic equations[J]. Applied Mathematics and Optimization, 1980, 6: 201-220. doi: 10.1007/BF01442895
|
| [18] |
ARGOMEDO F B, PRIEUR C, WITRANT E, et al. A strict control Lyapunov function for a diffusion equation with time-varying distributed coefficients[J]. IEEE Transactions on Automatic Control, 2013, 58(2): 290-303. doi: 10.1109/TAC.2012.2209260
|
| [19] |
LIONS J L. Exact controllability, stabilization and perturbations for distributed systems[J]. SIAM Review, 1988, 30(1): 1-68. doi: 10.1137/1030001
|
| [20] |
LIU W J, KRISTIC M. Backstepping boundary control of Burgers' equation with actuator dynamics[J]. Systems & Control Letters, 2000, 41(4): 291-303.
|
| [21] |
LIU H B, HU P, MUNTEANU I. Boundary feedback stabilization of Fisher's equation[J]. Systems & Control Le-tters, 2016, 97: 55-60.
|
| [22] |
BARBU V, LASIECKA I, TRIGGIANI R. Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high- and low-gain feedback contro-llers[J]. Nonlinear Analysis-Theory Methods & Applications, 2006, 64(12): 2704-2746.
|
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