Approximasi Thời Gian Bán Rời Đối Với Kiểm Soát Biên Dirichlet Cho Phương Trình Phát Triển Phân Số/Bình Thường Có Quan Sát Cuối Cùng

Springer Science and Business Media LLC - Tập 88 - Trang 1-28 - 2021
Qin Zhou1,2, Binjie Li1
1School of Mathematics, Sichuan University, Chengdu, China
2China West Normal University, Nanchong, China

Tóm tắt

Kiểm soát biên Dirichlet tối ưu cho một phương trình phát triển phân số/bình thường với quan sát cuối cùng được xem xét. Sự tồn tại duy nhất của nghiệm và điều kiện tối ưu bậc nhất của bài toán kiểm soát tối ưu được suy ra. Sự hội tụ của một phương pháp xấp xỉ bán rời theo thời gian được thiết lập một cách nghiêm ngặt, trong đó kiểm soát không được số hóa rõ ràng và phương trình trạng thái được số hóa bằng phương pháp Galerkin không liên tục theo thời gian. Các kết quả số được cung cấp để xác minh các kết quả lý thuyết.

Từ khóa

#Kiểm soát biên Dirichlet #phương trình phát triển phân số #phương pháp Galerkin không liên tục #xấp xỉ bán rời theo thời gian #điều kiện tối ưu.

Tài liệu tham khảo

Alt, W., Mackenroth, U.: Convergence of finite element approximations to state constrained convex parabolic boundary control problems. SIAM J. Control Optim. 27(4), 718–736 (1989) Arendt, W., Batty, C., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems, 2nd edn. Springer, Basel (2011) Antil, H., Otárola, E., Salgado, A.J.: A space-time fractional optimal control problem: analysis and discretization. SIAM J. Control Optim. 54(3), 1295–1328 (2016) Balakrishnan, A.V.: Applied Functional Analysis. Springer, New York (1981) Belgacem, F.B., Bernardi, C., Fekih, H.E.: Dirichlet boundary control for a parabolic equation with a final observation I: a space-time mixed formulation and penalization. Asymptot. Anal. 71, 101–121 (2011) Berggren, M.: Approximations of very weak solutions to boundary-value problems. SIAM J. Numer. Anal. 42(2), 860–877 (2004) Calderón, A.: Intermediate spaces and interpolation, the complex method. Stud. Math. 24(2), 113–190 (1964) Chrysafinos, K., Karatzas, E.N.: Error estimates for discontinuous Galerkin time-stepping schemes for Robin boundary control problems constrained to parabolic PDEs. SIAM J. Numer. Anal. 52(6), 2837–2862 (2014) Deckelnick, K., Hinze, M.: Variational discretization of parabolic control problems in the presence of pointwise state constraints. J. Comput. Math. 29, 1–15 (2011) Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010) Eriksson, K., Johnson, C., Thomée, V.: Time discretization of parabolic problems by the discontinuous Galerkin method. RAIRO Modél. Math. Anal. Numér. 19, 611–643 (1985) Ervin, V., Roop, J.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Meth. Part. D. E. 22(3), 558–576 (2006) Evans, L.C.: Partial Differential Equations, 2nd edn. American Mathematical Society, London (2010) French, D.A., King, J.T.: Approximation of an elliptic control problem by the finite element method. Numer. Funct. Anal. Optim. 12, 299–314 (1991) French, D.A., King, J.T.: Analysis of a robust finite element approximation for a parabolic equation with rough boundary data. Math. Comput. 60, 79–104 (1993) Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985) Gong, W., Hinze, M., Zhou, Z.: A priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control. SIAM J. Control Optim. 52, 97–119 (2014) Gong, W., Hinze, M., Zhou, Z.: Finite element method and a priori error estimates for Dirichlet boundary control problems governed by parabolic pdes. J. Sci. Comput. 66, 941–967 (2016) Gong, W., Li, B.: Improved error estimates for semidiscrete finite element solutions of parabolic Dirichlet boundary control problems. IMA J. Numer. Anal. 40(4), 2898–2939 (2020) Gunzburger, M., Wang, J.: Error analysis of fully discrete finite element approximations to an optimal control problem governed by a time-fractional pde. SIAM J. Control Optim. 57(1), 241–263 (2019) Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30, 45–63 (2005) Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, Netherlands (2009) Jin, B., Lazarov, R., Pasciak, J., Zhou, Z.: Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion. IMA J. Numer. Anal. 35, 561–582 (2015) Jin, B., Lazarov, R., Zhou, Z.: Error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM J. Numer. Anal. 51(1), 445–466 (2013) Jin, B., Lazarov, R., Zhou, Z.: An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36, 197–221 (2016) Jin, B., Lazarov, R., Zhou, Z.: Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Sci. Comput. 38(1), A146–A170 (2016) Jin, B., Li, B., Zhou, Z.: Discrete maximal regularity of time-stepping schemes for fractional evolution equations. Numer. Math. 138, 101–131 (2018) Jin, B., Li, B., Zhou, Z.: Subdiffusion with a time-dependent coefficient: analysis and numerical solution. Math. Comput. 88, 2157–2186 (2019) Jin, B., Li, B., Zhou, Z.: Pointwise-in-time error estimates for an optimal control problem with subdiffusioin constraint. IMA J. Numer. Anal. 40, 377–404 (2020) Jin, B., Li, B., Zhou, Z.: Numerical analysis of nonlinear subdiffusion equations. SIAM J. Numer. Anal. 56(1), 1–23 (2018) Knowles, G.: Finite element approximation of parabolic time optimal control problems. SIAM J. Control Optim. 20(3), 414–427 (1982) Kunisch, K., Vexler, B.: Constrained Dirichlet boundary control in \( L^2 \) for a class of evolution equations. SIAM J. Control Optim. 46, 1726–1753 (2007) Lasiecka, I.: Boundary control of parabolic systems: Finite-element approximation. Appl. Math. Optim. 31, 31–62 (1980) Lasiecka, I.: Unified theory for abstract parabolic boundary problems—a semigroup approach. Appl. Math. Optim. 6, 287–333 (1980) Lasiecka, I.: Ritz–Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions. SIAM J. Control Optim. 22(3), 477–500 (1984) Lasiecka, I.: Galerkin approximations of abstract parabolic boundary value problems with rough boundary data-\(L_p\) theory. Math. Comput. 47, 55–75 (1986) Lasiecka, I., Triggiani, R.: Dirichlet boundary control problem for parabolic equations with quadratic cost: Analyticity and riccati’s feedback synthesis. SIAM J. Control Optim. 21(1), 41–67 (1983) Leykekhman, D., Vexler, B.: Optimal a priori error estimates of parabolic optimal problems with pointwise control. SIAM J. Numer. Anal. 51, 2797–2821 (2013) Leykekhman, D., Vexler, B.: A priori error estimates for three dimensional parabolic optimal control problems with pointwise control. SIAM J. Numer. Anal. 54, 2403–2435 (2016) Li, B., Luo, H., Xie, X.: Analysis of a time-stepping scheme for time fractional diffusion problems with nonsmooth data. SIAM J. Numer. Anal. 57(2), 779–798 (2019) Li, B., Wang, T., Xie, X.: Numerical analysis of two Galerkin discretizations with graded temporal grids for fractional evolution equations. J. Sci. Comput. 85, 59 (2020) Li, B., Xie, X., Yan, Y.: L1 scheme for solving an inverse problem subject to a fractional diffusion equation. submitted, arXiv:2006.04291, (2020) Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225(2), 1533–1552 (2007) Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications, vol. 1. Springer, Berlin (1972) Lubich, C., Sloan, I., Thomée, V.: Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term. Math. Comput. 65(213), 1–17 (1996) Lunardi, A.: Interpolation Theory. Edizioni della Normale, Pisa (2018) Luo, H., Li, B., Xie, X.: Convergence analysis of a Petrov–Galerkin method for fractional wave problems with nonsmooth data. J. Sci. Comput. 80(2), 957–992 (2019) Malanowski, K.: Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems. Appl. Math. Optim. 8, 69–95 (1981) Mclean, W., Mustapha, K.: Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation. Numer. Algor. 52(1), 69–88 (2009) McLean, W., Thomée, V.: Numerical solution via Laplace transforms of a fractional order evolution equation. J. Integral Equ. Appl. 22, 57–94,03 (2010) Meidner, D., Vexler, B.: Adaptive space-time finite element methods for parabolic optimization problems. SIAM J. Control Optim. 46, 116–142 (2007) Meidner, D., Vexler, B.: A priori error estimates for space-time finite element discretization of parabolic optimal control problems part I: problems without control constraints. SIAM J. Control Optim. 47, 1150–1177 (2008) Meidner, D., Vexler, B.: A priori error estimates for space-time finite element discretization of parabolic optimal control problems part II: Problems with control constraints. SIAM J. Control Optim. 47(3), 1301–1329 (2008) Meidner, D., Vexler, B.: A priori error analysis of the Petrov–Galerkin Crank–Nicolson scheme for parabolic optimal control problems. SIAM J. Control Optim. 49(5), 2183–2211 (2011) Mustapha, K., Abdallah, B., Furati, K.: A discontinuous Petrov-Galerkin method for time-fractional diffusion equations. Fuel 58(12), 896–897 (2014) Mustapha, K., Mclean, W.: Discontinuous Galerkin method for an evolution equation with a memory term of positive type. Math. Comput. 78(268), 1975–1995 (2009) Mustapha, K., McLean, W.: Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation. Numer. Algor. 56(2), 159–184 (2011) Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983) Podlubny, I.: Fractional Differential Equations. Academic Press, Cambridge (1998) Sun, Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56(2), 193–209 (2006) Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006) Troltzsch, F.: Optimal Control of Partial Differential Equations. American Mathematical Society, Providence (2010) Yagi, A.: Abstract Parabolic Evolution Equations and their Applications. Springer, Berlin (2010) Zhang, C., Liu, H., Zhou, Z.: A priori error analysis for time-stepping discontinuous Galerkin finite element approximation of time fractional optimal control problem. J. Sci. Comput. 80, 993–1018 (2019)