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# Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

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Volume 18, Issue 1

# Sparse Optimal Control for Fractional Diffusion

Enrique Otárola
/ Abner J. Salgado
Published Online: 2017-09-02 | DOI: https://doi.org/10.1515/cmam-2017-0030

## 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.

MSC 2010: 26A33; 35J70; 49K20; 49M25; 65M12; 65M15; 65M60

## 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.

• [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.

• [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. 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.

• [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.

• [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.

• [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.

• [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.

• [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]

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.

• [11]

E. Casas, R. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with ${L}^{1}$ cost functional, SIAM J. Optim. 22 (2012), no. 3, 795–820.

• [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. Google Scholar

• [13]

P. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, Philadelphia, 2002. Google Scholar

• [14]

F. H. Clarke, Optimization and Nonsmooth Analysis, 2nd ed., Class. Appl. Math. 5, SIAM, Philadelphia, 1990. Google Scholar

• [15]

R. Durán and A. Lombardi, Error estimates on anisotropic ${Q}_{1}$ elements for functions in weighted Sobolev spaces, Math. Comp. 74 (2005), no. 252, 1679–1706. Google Scholar

• [16]

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Appl. Math. Sci. 159, Springer, New York, 2004. 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.

• [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.

• [19]

V. Gol’dshtein and A. Ukhlov, Weighted Sobolev spaces and embedding theorems, Trans. Amer. Math. Soc. 361 (2009), no. 7, 3829–3850.

• [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.

• [21]

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

• [22]

N. Landkof, Foundations of Modern Potential Theory, Grundlehren Math. Wiss. 180, Springer, Berlin, 1972. Google Scholar

• [23]

S. Levendorskiĭ, Pricing of the American put under Lévy processes, Int. J. Theor. Appl. Finance 7 (2004), no. 3, 303–335.

• [24]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer, New York, 1972. Google Scholar

• [25]

B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226.

• [26]

R. Musina and A. I. Nazarov, On fractional Laplacians, Comm. Partial Differential Equations 39 (2014), no. 9, 1780–1790.

• [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.

• [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.

• [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.

• [30]

W. Schirotzek, Nonsmooth Analysis, Universitext, Springer, Berlin, 2007. Google Scholar

• [31]

G. Stadler, Elliptic optimal control problems with ${L}^{1}$-control cost and applications for the placement of control devices, Comput. Optim. Appl. 44 (2009), no. 2, 159–181.

• [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.

• [33]

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

• [34]

F. Tröltzsch, Optimal Control of Partial Differential Equations, Grad. Stud. Math. 112, American Mathematical Society, Providence, 2010. Google Scholar

• [35]

B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Springer, Berlin, 2000. Google Scholar

• [36]

G. Vossen and H. Maurer, On ${L}^{1}$-minimization in optimal control and applications to robotics, Optimal Control Appl. Methods 27 (2006), no. 6, 301–321. 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.

## About the article

Revised: 2017-08-03

Accepted: 2017-08-05

Published Online: 2017-09-02

Published in Print: 2018-01-01

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1418784

Funding Source: Consejo Nacional de Innovación, Ciencia y Tecnología

Award identifier / Grant number: 3160201

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.

Citation Information: Computational Methods in Applied Mathematics, Volume 18, Issue 1, Pages 95–110, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840,

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© 2018 Walter de Gruyter GmbH, Berlin/Boston.

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