Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter September 2, 2017

Sparse Optimal Control for Fractional Diffusion

Enrique Otárola and Abner J. Salgado EMAIL logo

Abstract

We consider an optimal control problem that entails the minimization of a nondifferentiable cost functional, fractional diffusion as state equation and constraints on the control variable. We provide existence, uniqueness and regularity results together with first-order optimality conditions. In order to propose a solution technique, we realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic operator and consider an equivalent optimal control problem with a nonuniformly elliptic equation as state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We propose a fully discrete scheme: piecewise constant functions for the control variable and first-degree tensor product finite elements for the state variable. We derive a priori error estimates for the control and state variables.

Award Identifier / Grant number: DMS-1418784

Award Identifier / Grant number: 3160201

Funding statement: The first author has been supported in part by CONICYT through FONDECYT project 3160201. The second author has been supported in part by NSF grant DMS-1418784.

References

[1] H. Antil and E. Otárola, A FEM for an optimal control problem of fractional powers of elliptic operators, SIAM J. Control Optim. 53 (2015), no. 6, 3432–3456. 10.1137/140975061Search in Google Scholar

[2] H. Antil, E. Otárola and A. J. Salgado, A space-time fractional optimal control problem: Analysis and discretization, SIAM J. Control Optim. 54 (2016), no. 3, 1295–1328. 10.1137/15M1014991Search in Google Scholar

[3] T. Atanackovic, S. Pilipovic, B. Stankovic and D. Zorica, Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes, John Wiley & Sons, Hoboken, 2014. Search in Google Scholar

[4] L. Banjai, J. Melenk, R. H. Nochetto, E. Otárola, A. J. Salgado and C. Schwab, Tensor FEM for spectral fractional diffusion, preprint (2017), https://arxiv.org/abs/1707.07367. Search in Google Scholar

[5] A. Bonito, J. P. Borthagaray, R. H. Nochetto, E. Otárola and A. J. Salgado, Numerical methods for fractional diffusion, preprint (2017), https://arxiv.org/abs/1707.01566. Search in Google Scholar

[6] A. Bueno-Orovio, D. Kay, V. Grau, B. Rodriguez and K. Burrage, Fractional diffusion models of cardiac electrical propagation: Role of structural heterogeneity in dispersion of repolarization, J. R. Soc. Interface 11 (2014), no. 97. 10.1098/rsif.2014.0352Search in Google Scholar PubMed PubMed Central

[7] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), no. 5, 2052–2093. 10.1016/j.aim.2010.01.025Search in Google Scholar

[8] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7–9, 1245–1260. 10.1080/03605300600987306Search in Google Scholar

[9] A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations 36 (2011), no. 8, 1353–1384. 10.1080/03605302.2011.562954Search in Google Scholar

[10] E. Casas, R. Herzog and G. Wachsmuth, Approximation of sparse controls in semilinear equations by piecewise linear functions, Numer. Math. 122 (2012), no. 4, 645–669. 10.1007/s00211-012-0475-7Search in Google Scholar

[11] E. Casas, R. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with L1 cost functional, SIAM J. Optim. 22 (2012), no. 3, 795–820. 10.1137/110834366Search in Google Scholar

[12] W. Chen, A speculative study of 2/3-order fractional laplacian modeling of turbulence: Some thoughts and conjectures, Chaos 16 (2006), no. 2, 1–11. Search in Google Scholar

[13] P. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, Philadelphia, 2002. 10.1137/1.9780898719208Search in Google Scholar

[14] F. H. Clarke, Optimization and Nonsmooth Analysis, 2nd ed., Class. Appl. Math. 5, SIAM, Philadelphia, 1990. 10.1137/1.9781611971309Search in Google Scholar

[15] R. Durán and A. Lombardi, Error estimates on anisotropic Q1 elements for functions in weighted Sobolev spaces, Math. Comp. 74 (2005), no. 252, 1679–1706. 10.1090/S0025-5718-05-01732-1Search in Google Scholar

[16] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Appl. Math. Sci. 159, Springer, New York, 2004. 10.1007/978-1-4757-4355-5Search in Google Scholar

[17] D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad. 43 (1967), 82–86. 10.3792/pja/1195521686Search in Google Scholar

[18] P. Gatto and J. Hesthaven, Numerical approximation of the fractional Laplacian via hp-finite elements, with an application to image denoising, J. Sci. Comput. 65 (2015), no. 1, 249–270. 10.1007/s10915-014-9959-1Search in Google Scholar

[19] V. Gol’dshtein and A. Ukhlov, Weighted Sobolev spaces and embedding theorems, Trans. Amer. Math. Soc. 361 (2009), no. 7, 3829–3850. 10.1090/S0002-9947-09-04615-7Search in Google Scholar

[20] R. Ishizuka, S.-H. Chong and F. Hirata, An integral equation theory for inhomogeneous molecular fluids: The reference interaction site model approach, J. Chem. Phys. 128 (2008), no. 3. 10.1063/1.2819487Search in Google Scholar PubMed

[21] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Pure Appl. Math. 88, Academic Press, New York, 1980. Search in Google Scholar

[22] N. Landkof, Foundations of Modern Potential Theory, Grundlehren Math. Wiss. 180, Springer, Berlin, 1972. 10.1007/978-3-642-65183-0Search in Google Scholar

[23] S. Levendorskiĭ, Pricing of the American put under Lévy processes, Int. J. Theor. Appl. Finance 7 (2004), no. 3, 303–335. 10.1142/S0219024904002463Search in Google Scholar

[24] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer, New York, 1972. 10.1007/978-3-642-65161-8Search in Google Scholar

[25] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. 10.1090/S0002-9947-1972-0293384-6Search in Google Scholar

[26] R. Musina and A. I. Nazarov, On fractional Laplacians, Comm. Partial Differential Equations 39 (2014), no. 9, 1780–1790. 10.1080/03605302.2013.864304Search in Google Scholar

[27] R. H. Nochetto, E. Otárola and A. J. Salgado, A PDE approach to fractional diffusion in general domains: A priori error analysis, Found. Comput. Math. 15 (2015), no. 3, 733–791. 10.1007/s10208-014-9208-xSearch in Google Scholar

[28] R. H. Nochetto, E. Otárola and A. J. Salgado, A PDE approach to space-time fractional parabolic problems, SIAM J. Numer. Anal. 54 (2016), no. 2, 848–873. 10.1137/14096308XSearch in Google Scholar

[29] E. Otárola, A piecewise linear FEM for an optimal control problem of fractional operators: Error analysis on curved domains, ESAIM Math. Model. Numer. Anal. 51 (2017), no. 4, 1473–1500. 10.1051/m2an/2016065Search in Google Scholar

[30] W. Schirotzek, Nonsmooth Analysis, Universitext, Springer, Berlin, 2007. 10.1007/978-3-540-71333-3Search in Google Scholar

[31] G. Stadler, Elliptic optimal control problems with L1-control cost and applications for the placement of control devices, Comput. Optim. Appl. 44 (2009), no. 2, 159–181. 10.1007/s10589-007-9150-9Search in Google Scholar

[32] P. R. Stinga and J. L. Torrea, Extension problem and Harnack’s inequality for some fractional operators, Comm. Partial Differential Equations 35 (2010), no. 11, 2092–2122. 10.1080/03605301003735680Search in Google Scholar

[33] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lect. Notes Unione Mat. Ital. 3, Springer, Berlin, 2007. Search in Google Scholar

[34] F. Tröltzsch, Optimal Control of Partial Differential Equations, Grad. Stud. Math. 112, American Mathematical Society, Providence, 2010. 10.1090/gsm/112Search in Google Scholar

[35] B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Springer, Berlin, 2000. 10.1007/BFb0103908Search in Google Scholar

[36] G. Vossen and H. Maurer, On L1-minimization in optimal control and applications to robotics, Optimal Control Appl. Methods 27 (2006), no. 6, 301–321. 10.1002/oca.781Search in Google Scholar

[37] G. Wachsmuth and D. Wachsmuth, Convergence and regularization results for optimal control problems with sparsity functional, ESAIM Control Optim. Calc. Var. 17 (2011), no. 3, 858–886. 10.1051/cocv/2010027Search in Google Scholar

Received: 2017-4-4
Revised: 2017-8-3
Accepted: 2017-8-5
Published Online: 2017-9-2
Published in Print: 2018-1-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 8.12.2022 from frontend.live.degruyter.dgbricks.com/document/doi/10.1515/cmam-2017-0030/html
Scroll Up Arrow