Ước Lượng Sai Số Cho Kiểm Soát Tối Ưu Semilinear Phân Số Trên Các Đa Tọa Độ Lipschitz

Adams, R.A., Fournier, J.F.: Sobolev Spaces, second ed., vol. 140 of Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press, Amsterdam (2003) Antil, H., Otárola, E.: A FEM for an optimal control problem of fractional powers of elliptic operators. SIAM J. Control. Optim. 53(6), 3432–3456 (2015) Antil, H., Warma, M.: Optimal control of fractional semilinear PDEs. ESAIM Control Optim. Calc. Var. 26, 30 (2020) Antil, H., Khatri, R., Warma, M.: External optimal control of nonlocal PDEs. Inverse Probl. 8, 084003 (2019) Antil, H., Verma, D., Warma, M.: Optimal control of fractional elliptic PDEs with state constraints and characterization of the dual of fractional-order Sobolev spaces. J. Optim. Theory Appl. 186(1), 1–23 (2020) Bersetche, F., Fuica, F., Otárola, E., Quero, D.: Bilinear optimal control for the fractional laplacian: error estimates on lipschitz domains. arXiv:1605.03927 Bonito, A., Borthagaray, J.P., Nochetto, R.H., Otárola, E., Salgado, A.J.: Numerical methods for fractional diffusion. Comput. Vis. Sci. 19(5–6), 19–46 (2018) Borthagaray, J.P., Nochetto, R.H.: Besov regularity for the Dirichlet integral fractional Laplacian in Lipschitz domains. J. Funct. Anal. 284(6), 33 (2023) Borthagaray, J.P., Leykekhman, D., Nochetto, R.H.: Local energy estimates for the fractional Laplacian. SIAM J. Numer. Anal. 59(4), 1918–1947 (2021) Burkovska, O., Glusa, C., D’Elia, M.: An optimization-based approach to parameter learning for fractional type nonlocal models. Comput. Math. Appl. 116, 229–244 (2022) Chiappinelli, R., Nugari, R.: The Nemitskiĭ operator in Hölder spaces: some necessary and sufficient conditions. J. Lond. Math. Soc. 251(2), 365–372 (1995) D’Elia, M., Gunzburger, M.: Identification of the diffusion parameter in nonlocal steady diffusion problems. Appl. Math. Optim. 73(2), 227–249 (2016) D’Elia, M., Glusa, C., Otárola, E.: A priori error estimates for the optimal control of the integral fractional Laplacian. SIAM J. Control. Optim. 57(4), 2775–2798 (2019) D’Elia, M., Du, Q., Glusa, C., Gunzburger, M., Tian, X., Zhou, Z.: Numerical methods for nonlocal and fractional models. Acta Numer. 29, 1–124 (2020) Dohr, S., Kahle, C., Rogovs, S., Swierczynski, P.: A FEM for an optimal control problem subject to the fractional Laplace equation. Calcolo 56(4), 21 (2019) Faermann, B.: Localization of the Aronszajn–Slobodeckij norm and application to adaptive boundary element methods. I. The two-dimensional case. IMA J. Numer. Anal. 20(2), 203–234 (2000) Faermann, B.: Localization of the Aronszajn–Slobodeckij norm and application to adaptive boundary element methods. II. The three-dimensional case. Numer. Math. 92(3), 467–499 (2002) Getoor, R.K.: First passage times for symmetric stable processes in space. Trans. Am. Math. Soc. 101, 75–90 (1961) Gimperlein, H., Stephan, E., Stocek, J.: Corner singularities for the fractional Laplacian and finite element approximation. (2019) Glusa, C., Otárola, E.: Error estimates for the optimal control of a parabolic fractional PDE. SIAM J. Numer. Anal. 59(2), 1140–1165 (2021) Grubb, G.: Regularity of spectral fractional Dirichlet and Neumann problems. Math. Nachr. 289(7), 831–844 (2016) Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30(1), 45–61 (2005) Holler, G., Kunisch, K.: Learning nonlocal regularization operators. Math. Control Relat. Fields 12(1), 81–114 (2022) Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications, vol. 31 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000) Kwaśnicki, M.: Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal. 20(1), 7–51 (2017) Landkof, N.S.: Foundations of Modern Potential Theory. Springer, New York (1972) McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000) Oswald, P.: On the boundedness of the mapping \(f\rightarrow |f|\) in Besov spaces. Comment. Math. Univ. Carolin. 33(1), 57–66 (1992) Otárola, E.: A piecewise linear FEM for an optimal control problem of fractional operators: error analysis on curved domains. ESAIM Math. Model. Numer. Anal. 51(4), 1473–1500 (2017) Otárola, E.: An adaptive finite element method for the sparse optimal control of fractional diffusion. Numer. Methods Partial Differ. Equ. 36(2), 302–328 (2020) Otárola, E.: Fractional semilinear optimal control: optimality conditions, convergence, and error analysis. SIAM J. Numer. Anal. 60(1), 1–27 (2022) Otárola, E., Quyen, T.N.T.: A reaction coefficient identification problem for fractional diffusion. Inverse Probl. 35(4), 045010 (2019) Otárola, E., Salgado, A.J.: Sparse optimal control for fractional diffusion. Comput. Methods Appl. Math. 18(1), 95–110 (2018) Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. (9) 101(3), 275–302 (2014) Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces, vol. 3 of Lecture Notes of the Unione Matematica Italiana. Springer, Berlin (2007) Tröltzsch, F.: Optimal Control of Partial Differential Equations. Graduate Studies in Mathematics, vol. 112. American Mathematical Society, Providence, RI (2010)