Abstract
This paper introduces the notion of state constraints for optimal control problems governed by fractional elliptic partial differential equations. Several mathematical tools are developed during the process to study these problems, for instance, the characterization of the dual of fractional-order Sobolev spaces and the well-posedness of fractional elliptic equations with measure-valued data. These tools are widely applicable. We show well-posedness of the optimal control problem and derive the first- order optimality conditions. Notice that the adjoint equation is a fractional partial differential equation with a measure as the right-hand-side datum. We use the characterization of the fractional- order dual spaces to study the regularity of solutions of the state and adjoint equations. We emphasize that the classical case was considered by E. Casas, but almost none of the existing results are applicable to our fractional case. As an application of the regularity result of the adjoint equation, we establish the Sobolev regularity of the optimal control. In addition, under this setup, even weaker controls can be used.
Similar content being viewed by others
References
Weiss, C., van Bloemen Waanders, B., Antil, H.: Fractional operators applied to geophysical electromagnetics. Geophys. J. Int. 220(2), 1242–1259 (2020)
Antil, H., Rautenberg, C.: Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications. SIAM J. Math. Anal. 51(3), 2479–2503 (2019). https://doi.org/10.1137/18M1224970
Antil, H., Warma, M.: Optimal control of fractional semilinear PDE. ESAIM Control Optim. Calc. Var. 26, 30 (2020). https://doi.org/10.1051/cocv/2019003
D’Elia, M., Gunzburger, M.: Optimal distributed control of nonlocal steady diffusion problems. SIAM J. Control Optim. 52(1), 243–273 (2014). https://doi.org/10.1137/120897857
Antil, H., Pfefferer, J., Rogovs, S.: Fractional operators with inhomogeneous boundary conditions: analysis, control, and discretization. Commun. Math. Sci. 16(5), 1395–1426 (2018). https://doi.org/10.4310/CMS.2018.v16.n5.a11
Antil, H., Khatri, R., Warma, M.: External optimal control of nonlocal PDEs. Inverse Probl. 35(8), 084,003, 35 (2019)
Antil, H., Verma, D., Warma, M.: External optimal control of fractional parabolic PDEs. ESAIM Control Optim. Calc. Var. 26, (2020). https://doi.org/10.1051/cocv/2020005
Antil, H., Warma, M.: Optimal control of the coefficient for the regional fractional \(p\)-Laplace equation: approximation and convergence. Math. Control Relat. Fields 9(1), 1–38 (2019)
Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 24(6), 1309–1318 (1986). https://doi.org/10.1137/0324078
Casas, E.: Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31(4), 993–1006 (1993). https://doi.org/10.1137/0331044
Casas, E., Mateos, M., Vexler, B.: New regularity results and improved error estimates for optimal control problems with state constraints. ESAIM Control Optim. Calc. Var. 20(3), 803–822 (2014). https://doi.org/10.1051/cocv/2013084
Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE constraints. In: Mathematical Modelling: Theory and Applications, vol. 23. Springer, New York (2009)
Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Graduate Studies in Mathematics, vol. 112. American Mathematical Society, Providence, RI (2010). https://doi.org/10.1090/gsm/112
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). https://doi.org/10.1016/j.matpur.2013.06.003
Warma, M.: The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets. Potential Anal. 42(2), 499–547 (2015). https://doi.org/10.1007/s11118-014-9443-4
Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, vol. 24. Pitman (Advanced Publishing Program), Boston, MA (1985)
Bogdan, K., Burdzy, K., Chen, Z.Q.: Censored stable processes. Probab. Theory Relat. Fields 127(1), 89–152 (2003). https://doi.org/10.1007/s00440-003-0275-1
Fiscella, A., Servadei, R., Valdinoci, E.: Density properties for fractional Sobolev spaces. Ann. Acad. Sci. Fenn. Math. 40(1), 235–253 (2015). https://doi.org/10.5186/aasfm.2015.4009
Borthagaray, J., Ciarlet Jr., P.: On the convergence in \(H^1\)-norm for the fractional Laplacian. SIAM J. Numer. Anal. 57(4), 1723–1743 (2019). https://doi.org/10.1137/18M1221436
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012). https://doi.org/10.1016/j.bulsci.2011.12.004
Dipierro, S., Ros-Oton, X., Valdinoci, E.: Nonlocal problems with Neumann boundary conditions. Rev. Mat. Iberoam. 33(2), 377–416 (2017). https://doi.org/10.4171/RMI/942
Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization. MOS-SIAM Series on Optimization, vol. 17, 2nd edn. Society for Industrial and Applied Mathematic, Philadelphia (2014). https://doi.org/10.1137/1.9781611973488
Alt, H.W.: Linear Functional Analysis: An Application Oriented Introduction. Springer, Berlin (1992)
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)
Antil, H., Pfefferer, J., Warma, M.: A note on semilinear fractional elliptic equation: analysis and discretization. ESAIM: M2AN 51(6), 2049–2067 (2017). https://doi.org/10.1051/m2an/2017023
Antil, H., Kouri, D., Lacasse, M.D., Ridzal, D. (eds.): Frontiers in PDE-Constrained Optimization. The IMA Volumes in Mathematics and Its Applications, vol. 163. Springer, New York (2018). https://doi.org/10.1007/978-1-4939-8636-1
Grubb, G.: Fractional Laplacians on domains: a development of Hörmander’s theory of \(\mu \)-transmission pseudodifferential operators. Adv. Math. 268, 478–528 (2015). https://doi.org/10.1016/j.aim.2014.09.018
Grubb, G.: Local and nonlocal boundary conditions for \(\mu \)-transmission and fractional elliptic pseudodifferential operators. Anal. PDE 7(7), 1649–1682 (2014). https://doi.org/10.2140/apde.2014.7.1649
Antil, H., Brown, T., Verma, D.: Moreau-yosida regularization for optimal control of fractional elliptic problems with state and control constraints. arXiv preprint arXiv:1912.05033 (2019)
Acknowledgements
The first and second authors are partially supported by NSF Grants DMS-1818772 and DMS-1913004, the Air Force Office of Scientific Research under Award No.: FA9550-19-1-0036, and the Department of Navy, Naval PostGraduate School under Award No.: N00244-20-1-0005. The third author is partially supported by the Air Force Office of Scientific Research under Award No.: FA9550-18-1-0242.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Irena Lasiecka.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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, 1–23 (2020). https://doi.org/10.1007/s10957-020-01684-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-020-01684-z
Keywords
- Optimal control with PDE constraint
- State and control constraints
- Fractional Laplacian
- Measure valued data
- Characterization of fractional dual spaces
- Regularity of solutions to state and adjoint equations
- Regularity of optimal control