    March  2019, 9(1): 1-38. doi: 10.3934/mcrf.2019001

## Optimal control of the coefficient for the regional fractional $p$-Laplace equation: Approximation and convergence

 1 Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA 2 University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, College of Natural Sciences, 17 University AVE. STE 1701, San Juan PR 00925-2537, USA

* Corresponding author: Harbir Antil

Received  April 2017 Revised  March 2018 Published  August 2018

Fund Project: The work of the first author is partially supported by NSF grant DMS-1521590. The work of the second author is partially supported by the Air Force Office of Scientific Research (AFOSR) under the Award No: FA9550-15-1-0027.

In this paper we study optimal control problems with the regional fractional $p$-Laplace equation, of order $s \in \left( {0,1} \right)$ and $p \in \left[ {2,\infty } \right)$, as constraints over a bounded open set with Lipschitz continuous boundary. The control, which fulfills the pointwise box constraints, is given by the coefficient of the regional fractional $p$-Laplace operator. We show existence and uniqueness of solutions to the state equations and existence of solutions to the optimal control problems. We prove that the regional fractional $p$-Laplacian approaches the standard $p$-Laplacian as $s$ approaches 1. In this sense, this fractional $p$-Laplacian can be considered degenerate like the standard $p$-Laplacian. To overcome this degeneracy, we introduce a regularization for the regional fractional $p$-Laplacian. We show existence and uniqueness of solutions to the regularized state equation and existence of solutions to the regularized optimal control problem. We also prove several auxiliary results for the regularized problem which are of independent interest. We conclude with the convergence of the regularized solutions.

Citation: Harbir Antil, Mahamadi Warma. Optimal control of the coefficient for the regional fractional $p$-Laplace equation: Approximation and convergence. Mathematical Control & Related Fields, 2019, 9 (1) : 1-38. doi: 10.3934/mcrf.2019001
##### References:
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Pisani, Solitons in several space dimensions:Derrick's problem and infinitely many solutions, Arch. Ration. Mech. Anal., 154 (2000), 297-324.  doi: 10.1007/s002050000101. Google Scholar  C. Bjorland, L. Caffarelli and A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Comm. Pure Appl. Math., 65 (2012), 337-380.  doi: 10.1002/cpa.21379. Google Scholar  K. Bogdan, K. Burdzy and Z.-Q. Chen, Censored stable processes, Probab. Theory Related Fields, 127 (2003), 89-152.  doi: 10.1007/s00440-003-0275-1. Google Scholar  J. Bourgain, H. Brezis and P. Mironescu, Another look at sobolev spaces, in Optimal Control and Partial Differential Equation, Conference, 2001,439-455. Google Scholar  J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for $W^{s,p}$ when $s\uparrow1$ and applications, J. Anal. Math., 87 (2002), 77-101, Dedicated to the memory of Thomas H.  doi: 10.1007/BF02868470. Google Scholar  L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813. Google Scholar  L. Caffarelli, Non-local diffusions, drifts and game, Nonlinear Partial Differential Equations, Abel Sym- posia, 7 (2012), 37-52.   Google Scholar  L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. Google Scholar  L. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS), 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226. Google Scholar  L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.  doi: 10.1007/s00222-007-0086-6. Google Scholar  E. Casas, P. Kogut and G. Leugering, Approximation of optimal control problems in the coefficient for the $p$-Laplace equation. I. Convergence result, SIAM J. Control Optim., 54 (2016), 1406-1422.  doi: 10.1137/15M1028108. Google Scholar  A. Di Castro., T. Kuusi. and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836.  doi: 10.1016/j.jfa.2014.05.023. Google Scholar  A. Di Castro., T. Kuusi and G. Palatucci, Local behavior of fractional $p$-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299.  doi: 10.1016/j.anihpc.2015.04.003. Google Scholar  E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. Google Scholar  E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2. Google Scholar  L. Diening, A. Prohl and M. Růžička, Semi-implicit Euler scheme for generalized Newtonian fluids, SIAM J. Numer. Anal., 44 (2006), 1172-1190 (electronic).  doi: 10.1137/050634335. Google Scholar  L. Diening and S. Schwarzacher, Global gradient estimates for the $p$(·)-Laplacian, Nonlinear Anal., 106 (2014), 70-85.  doi: 10.1016/j.na.2014.04.006. Google Scholar  P. Drábek and J. Milota, Methods of Nonlinear Analysis, 2nd edition, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser/Springer Basel AG, Basel, 2013, Applications to differential equations. doi: 10.1007/978-3-0348-0387-8. Google Scholar  A. Elmoataz, M. Toutain and D. Tenbrinck, On the $p$-Laplacian and ∞-Laplacian on graphs with applications in image and data processing, SIAM J. Imaging Sci., 8 (2015), 2412-2451.  doi: 10.1137/15M1022793. Google Scholar  L. Evans, Partial differential equations and Monge-Kantorovich mass transfer, in Current Developments in Mathematics, 1997 (Cambridge, MA), Int. Press, Boston, MA, 1999, 65-126. Google Scholar  C.G. Gal and M. Warma, On some degenerate non-local parabolic equation associated with the fractional $p$-Laplacian, Dyn. Partial Differ. Equ., 14 (2017), 47-77.  doi: 10.4310/DPDE.2017.v14.n1.a4. Google Scholar  P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. Google Scholar  A. Jonsson and H. Wallin, Function spaces on subsets of ${\bf R}^n$, Math. Rep., 2 (1984), xiv+221pp. Google Scholar  O. Kupenko and R. Manzo, Approximation of an optimal control problem in coefficient for variational inequality with anisotropic p-Laplacian, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 35, 18pp. doi: 10.1007/s00030-016-0387-9. Google Scholar  T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317-1368.  doi: 10.1007/s00220-015-2356-2. Google Scholar  J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181. Google Scholar  F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010, An introduction to mathematical models. doi: 10.1142/9781848163300. Google Scholar  V. Maz'ya and S. Poborchi, Differentiable Functions on Bad Domains, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. Google Scholar  F. Murat, Un contre-exemple pour le problème du contrôle dans les coefficients, C. R. Acad. Sci. Paris Sér. A-B, 273 (1971), A708-A711.   Google Scholar  F. Murat, Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients, Ann. Mat. Pura Appl. (4), 112 (1977), 49-68.   Google Scholar  F. Murat and L. Tartar, H-convergence, in Topics in the Mathematical Modelling of Composite Materials, vol. 31 of Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 1997, 21-43. Google Scholar  I. Pan and S. Das, Intelligent Fractional Order Systems and Control: An Introduction, vol. 438, Springer, 2012. Google Scholar  M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, vol. 1748 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029. Google Scholar  R. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, vol. 49 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997. Google Scholar  L. Tartar, Problèmes de contrôle des coefficients dans des équations aux dérivées partielles, in Control Theory, Numerical Methods and Computer Systems Modelling (Internat. Sympos., IRIA LABORIA, Rocquencourt, 1974), Springer, Berlin, 1975,420-426. Lecture Notes in Econom. and Math. Systems, Vol. 107. Google Scholar  D. Valério and J. Sá da Costa, An Introduction to Fractional Control, vol. 91 of IET Control Engineering Series, Institution of Engineering and Technology (IET), London, 2013. Google Scholar  J. Vázquez, The Dirichlet problem for the fractional $p$-Laplacian evolution equation, J. Differential Equations, 260 (2016), 6038-6056.  doi: 10.1016/j.jde.2015.12.033. Google Scholar  M. Warma, A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains, Commun. Pure Appl. Anal., 14 (2015), 2043-2067.  doi: 10.3934/cpaa.2015.14.2043. Google Scholar  M. Warma, The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets, Potential Anal., 42 (2015), 499-547.  doi: 10.1007/s11118-014-9443-4. Google Scholar  M. Warma, The fractional Neumann and Robin type boundary conditions for the regional fractional p-Laplacian, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 1, 46. doi: 10.1007/s00030-016-0354-5. Google Scholar  M. Warma, Local Lipschitz continuity of the inverse of the fractional p-Laplacian, Hölder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains, Nonlinear Anal., 135 (2016), 129-157.  doi: 10.1016/j.na.2016.01.022. Google Scholar  M. Warma, On a fractional (s, p)-Dirichlet-to-Neumann operator on bounded lipschitz domains, J. Elliptic and Parabol. Equ., 4 (2018), 223-269.  doi: 10.1007/s41808-018-0017-2. Google Scholar

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##### References:
  D. Adams and L. Hedberg, Function Spaces and Potential Theory, vol. 314 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03282-4. Google Scholar  L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. Google Scholar  H. Antil and S. Bartels, Spectral approximation of fractional PDEs in image processing and phase field modeling, Comput. Methods Appl. Math., 17 (2017), 661-678.  doi: 10.1515/cmam-2017-0039. Google Scholar  H. Antil, J. Pfefferer and M. Warma, A note on semilinear fractional elliptic equation: analysis and discretization, Math. Model. Numer. Anal. (ESAIM: M2AN), 51 (2017), 2049-2067.   Google Scholar  V. Benci, P. D'Avenia, D. Fortunato and L. Pisani, Solitons in several space dimensions:Derrick's problem and infinitely many solutions, Arch. Ration. Mech. Anal., 154 (2000), 297-324.  doi: 10.1007/s002050000101. Google Scholar  C. Bjorland, L. Caffarelli and A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Comm. Pure Appl. Math., 65 (2012), 337-380.  doi: 10.1002/cpa.21379. Google Scholar  K. Bogdan, K. Burdzy and Z.-Q. Chen, Censored stable processes, Probab. Theory Related Fields, 127 (2003), 89-152.  doi: 10.1007/s00440-003-0275-1. Google Scholar  J. Bourgain, H. Brezis and P. Mironescu, Another look at sobolev spaces, in Optimal Control and Partial Differential Equation, Conference, 2001,439-455. Google Scholar  J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for $W^{s,p}$ when $s\uparrow1$ and applications, J. Anal. Math., 87 (2002), 77-101, Dedicated to the memory of Thomas H.  doi: 10.1007/BF02868470. Google Scholar  L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813. Google Scholar  L. Caffarelli, Non-local diffusions, drifts and game, Nonlinear Partial Differential Equations, Abel Sym- posia, 7 (2012), 37-52.   Google Scholar  L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. Google Scholar  L. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS), 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226. Google Scholar  L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.  doi: 10.1007/s00222-007-0086-6. Google Scholar  E. Casas, P. Kogut and G. Leugering, Approximation of optimal control problems in the coefficient for the $p$-Laplace equation. I. Convergence result, SIAM J. Control Optim., 54 (2016), 1406-1422.  doi: 10.1137/15M1028108. Google Scholar  A. Di Castro., T. Kuusi. and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836.  doi: 10.1016/j.jfa.2014.05.023. Google Scholar  A. Di Castro., T. Kuusi and G. Palatucci, Local behavior of fractional $p$-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299.  doi: 10.1016/j.anihpc.2015.04.003. Google Scholar  E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. Google Scholar  E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2. Google Scholar  L. Diening, A. Prohl and M. Růžička, Semi-implicit Euler scheme for generalized Newtonian fluids, SIAM J. Numer. Anal., 44 (2006), 1172-1190 (electronic).  doi: 10.1137/050634335. Google Scholar  L. Diening and S. Schwarzacher, Global gradient estimates for the $p$(·)-Laplacian, Nonlinear Anal., 106 (2014), 70-85.  doi: 10.1016/j.na.2014.04.006. Google Scholar  P. Drábek and J. Milota, Methods of Nonlinear Analysis, 2nd edition, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser/Springer Basel AG, Basel, 2013, Applications to differential equations. doi: 10.1007/978-3-0348-0387-8. Google Scholar  A. Elmoataz, M. Toutain and D. Tenbrinck, On the $p$-Laplacian and ∞-Laplacian on graphs with applications in image and data processing, SIAM J. Imaging Sci., 8 (2015), 2412-2451.  doi: 10.1137/15M1022793. Google Scholar  L. Evans, Partial differential equations and Monge-Kantorovich mass transfer, in Current Developments in Mathematics, 1997 (Cambridge, MA), Int. Press, Boston, MA, 1999, 65-126. Google Scholar  C.G. Gal and M. Warma, On some degenerate non-local parabolic equation associated with the fractional $p$-Laplacian, Dyn. Partial Differ. Equ., 14 (2017), 47-77.  doi: 10.4310/DPDE.2017.v14.n1.a4. Google Scholar  P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. Google Scholar  A. Jonsson and H. Wallin, Function spaces on subsets of ${\bf R}^n$, Math. Rep., 2 (1984), xiv+221pp. Google Scholar  O. Kupenko and R. Manzo, Approximation of an optimal control problem in coefficient for variational inequality with anisotropic p-Laplacian, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 35, 18pp. doi: 10.1007/s00030-016-0387-9. Google Scholar  T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317-1368.  doi: 10.1007/s00220-015-2356-2. Google Scholar  J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181. Google Scholar  F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010, An introduction to mathematical models. doi: 10.1142/9781848163300. Google Scholar  V. Maz'ya and S. Poborchi, Differentiable Functions on Bad Domains, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. Google Scholar  F. Murat, Un contre-exemple pour le problème du contrôle dans les coefficients, C. R. Acad. Sci. Paris Sér. A-B, 273 (1971), A708-A711.   Google Scholar  F. Murat, Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients, Ann. Mat. Pura Appl. (4), 112 (1977), 49-68.   Google Scholar  F. Murat and L. Tartar, H-convergence, in Topics in the Mathematical Modelling of Composite Materials, vol. 31 of Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 1997, 21-43. Google Scholar  I. Pan and S. Das, Intelligent Fractional Order Systems and Control: An Introduction, vol. 438, Springer, 2012. Google Scholar  M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, vol. 1748 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029. Google Scholar  R. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, vol. 49 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997. Google Scholar  L. Tartar, Problèmes de contrôle des coefficients dans des équations aux dérivées partielles, in Control Theory, Numerical Methods and Computer Systems Modelling (Internat. Sympos., IRIA LABORIA, Rocquencourt, 1974), Springer, Berlin, 1975,420-426. Lecture Notes in Econom. and Math. Systems, Vol. 107. Google Scholar  D. Valério and J. Sá da Costa, An Introduction to Fractional Control, vol. 91 of IET Control Engineering Series, Institution of Engineering and Technology (IET), London, 2013. Google Scholar  J. Vázquez, The Dirichlet problem for the fractional $p$-Laplacian evolution equation, J. Differential Equations, 260 (2016), 6038-6056.  doi: 10.1016/j.jde.2015.12.033. Google Scholar  M. Warma, A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains, Commun. Pure Appl. Anal., 14 (2015), 2043-2067.  doi: 10.3934/cpaa.2015.14.2043. Google Scholar  M. Warma, The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets, Potential Anal., 42 (2015), 499-547.  doi: 10.1007/s11118-014-9443-4. Google Scholar  M. Warma, The fractional Neumann and Robin type boundary conditions for the regional fractional p-Laplacian, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 1, 46. doi: 10.1007/s00030-016-0354-5. Google Scholar  M. Warma, Local Lipschitz continuity of the inverse of the fractional p-Laplacian, Hölder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains, Nonlinear Anal., 135 (2016), 129-157.  doi: 10.1016/j.na.2016.01.022. Google Scholar  M. Warma, On a fractional (s, p)-Dirichlet-to-Neumann operator on bounded lipschitz domains, J. Elliptic and Parabol. Equ., 4 (2018), 223-269.  doi: 10.1007/s41808-018-0017-2. Google Scholar
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