一类非线性抛物最优控制问题的Crank-Nicolson有限元近似格式误差分析
Error Estimate for the Crank-Nicolson Finite Element Scheme Approximation of Nonlinear Parabolic Optimal Control Problems
DOI: 10.12677/ORF.2023.134415, PDF, 下载: 136  浏览: 215  国家自然科学基金支持
作者: 丁美玲, 李欢欢, 罗贤兵*：贵州大学数学与统计学院，贵州 贵阳
关键词: 非线性；抛物最优控制；Crank-Nicolson 格式；先验误差估计；有限元；Nonlinear； Parabolic Optimal Control； Crank-Nicolson Scheme； Priori Error Estimates； Finite Element
摘要: 本文对一类非线性抛物最优控制问题给出了Crank-Nicolson有限元近似格式。对于状态y和伴随状态变量p采用线性协调有限元离散，对控制变量u采用分片常数近似；得到了控制和状态变量近似的先验误差估计O(h + hu + (∆t)2)，为验证算法的有效性给出了数值算例。
Abstract: In this paper, we present a Crank − Nicolson scheme combined with finite element approximation for a class of nonlinear parabolic optimal control problems, where the state y and co-state p are discretized by picewise linear continuous function and the control u is approximations by picewise constant functions. Numerical experiments verify that the convergence order is O(h + hU + (∆t)2).
文章引用：丁美玲, 李欢欢, 罗贤兵. 一类非线性抛物最优控制问题的Crank-Nicolson有限元近似格式误差分析[J]. 运筹与模糊学, 2023, 13(4): 4147-4166. https://doi.org/10.12677/ORF.2023.134415

参考文献

[1] Liu, W.B. and Yan, N.N. (2002) A Posteriori Error Estimates for Control Problems Governed by Stockes Equations. SIAM Journal on Numerical Analysis, 40, 1850-1869.
https://doi.org/10.1137/S0036142901384009
[2] Neitzel, I. and Vexler, B. (2012) A Priori Error Estimates for Space-Time Finite Element Discretization of Semilinear Parabolic Optimal Control Problems. Numerische Mathematik, 120, 345-386.
https://doi.org/10.1007/s00211-011-0409-9
[3] 付红斐, 芮洪兴. 一类非线性抛物最优控制问题的有限元误差估计[J]. 数学物理学报, 2010,30(6): 1542-1554.
[4] Chatzipantelidis, P., Lazarov, R.D. and Thom´ee, V. (2004) Error Estimates for a Finite Volume Element Method for Parabolic Equations in Convex Polygonal Domains. Numerical Methods for Partial Differential Equations: An International Journal, 20, 650-674.
https://doi.org/10.1002/num.20006
[5] Marburger, J. and Pinnau, R. (2013) Optimal Control for Burgers Equation Using Particle Methods. arXiv:1309.7619
[6] Burman, E. (2011) Crank-Nicolson Finite Element Methods Using Symmetric Stabilization with an Application to Optimal Control Problems Subject to Transient Advection-Diffusion Equations. Communications in Mathematical Sciences, 9, 319-329.
https://doi.org/10.4310/CMS.2011.v9.n1.a16
[7] Chen, Y. and Lu, Z. (2010) Error Estimates of Fully Discrete Mixed Finite Element Methods for Semilinear Quadratic Parabolic Optimal Control Problem. Computer Methods in Applied Mechanics and Engineering, 199, 1415-1423.
https://doi.org/10.1016/j.cma.2009.11.009
[8] Wang, J., Lu, Z., Cai, F., et al. (2022) Fully Discrete Interpolation Coefficients Mixed Finite Element Methods for Semi-Linear Parabolic Optimal Control Problem. IEEE Access, 10, 54291-54300.
https://doi.org/10.1109/ACCESS.2022.3173402
[9] Hou, C., Lu, Z., Chen, X., et al. (2021) Error Estimates of Variational Discretization for Semilinear Parabolic Optimal Control Problems. AIMS Mathematics, 6, 772-793.
https://doi.org/10.3934/math.2021047
[10] Yang, C. and Sun, T. (2022) Crank-Nicolson Finite Difference Schemes for Parabolic Optimal Dirichlet Boundary Control Problems. Mathematical Methods in the Applied Sciences, 45, 7346-7363.
https://doi.org/10.1002/mma.8244
[11] Tang, Y. (2023) Error Estimates of Mixed Finite Elements Combined with Crank-Nicolson Scheme for Parabolic Control Problems. AIMS Mathematics, 8, 12506-12519.
https://doi.org/10.3934/math.2023628
[12] Lions, J.L. (1971) Optimal Control of Systems Governed by Partial Differential Equations.Springer-Verlag, Berlin.
[13] Neittaanmaki, P. and Tiba, D. (1994) Optimal Control of Nonlinear Parabolic Systems—Theory, Algorithms and Applications. Marcel Dekker, New York.
[14] Wheeler, M.F. (1973) A Priori L2 Error Estimates for Galerkin Approximations to Parabolic Partial Differential Equations. SIAM Journal on Numerical Analysis, 10, 723-759.
https://doi.org/10.1137/0710062
[15] Ciarlet, P.G. (2002) The Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics, Philadelphia.
https://doi.org/10.1137/1.9780898719208

为你推荐