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doi: 10.3934/eect.2021014

Internal control for a non-local Schrödinger equation involving the fractional Laplace operator

1. 

Chair of Computational Mathematics, Fundación Deusto, Avda. de las Universidades 24, 48007 Bilbao, Basque Country, Spain

2. 

Facultad de Ingeniería, Universidad de Deusto, Avda. de las Universidades 24, 48007 Bilbao, Basque Country, Spain

Received  October 2020 Published  April 2021

Fund Project: This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement NO: 694126-DyCon). This work was partially supported by the Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain), by the Elkartek grant KK-2020/00091 CONVADP of the Basque government and by the Grant FA9550-18-1-0242 of AFOSR

We analyze the interior controllability problem for a non-local Schrödinger equation involving the fractional Laplace operator $ (-\Delta)^{\, {s}}{} $, $ s\in(0, 1) $, on a bounded $ C^{1, 1} $ domain $ \Omega\subset{\mathbb{R}}^N $. We first consider the problem in one space dimension and employ spectral techniques to prove that, for $ s\in[1/2, 1) $, null-controllability is achieved through an $ L^2(\omega\times(0, T)) $ function acting in a subset $ \omega\subset\Omega $ of the domain. This result is then extended to the multi-dimensional case by applying the classical multiplier method, joint with a Pohozaev-type identity for the fractional Laplacian.

Citation: Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations & Control Theory, doi: 10.3934/eect.2021014
References:
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H. Antil, U. Biccari, R. Ponce, M. Warma and S. Zamorano, Controllability properties from the exterior under positivity constraints for a 1-d fractional heat equation, arXiv preprint, arXiv: 1910.14529. Google Scholar

[2]

H. Antil, R. Khatri and M. Warma, External optimal control of nonlocal PDEs, Inverse Problems, 35 (2019), 084003, 35 pp. doi: 10.1088/1361-6420/ab1299.  Google Scholar

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C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.  Google Scholar

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J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Berlin, 1976. Google Scholar

[5]

U. Biccari and V. Hernández-Santamarıa, Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects, IMA J. Math. Control Inf., 36 (2019), 1199-1235.  doi: 10.1093/imamci/dny025.  Google Scholar

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U. Biccari and M. Warma, Null-controllability properties of a fractional wave equation with a memory term, Evol. Eq. Control. Theo., 9 (2020), 399-430.  doi: 10.3934/eect.2020011.  Google Scholar

[7]

U. BiccariM. Warma and E. Zuazua, Addendum: Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlin. Stud., 17 (2017), 837-839.  doi: 10.1515/ans-2017-6020.  Google Scholar

[8]

U. BiccariM. Warma and E. Zuazua, Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlin. Stud., 17 (2017), 387-409.  doi: 10.1515/ans-2017-0014.  Google Scholar

[9]

U. BiccariM. Warma and E. Zuazua, Controllability of the one-dimensional fractional heat equation under positivity constraints, Commun. Pure Appl. Anal., 19 (2020), 1949-1980.  doi: 10.3934/cpaa.2020086.  Google Scholar

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B. Claus and M. Warma, Realization of the fractional Laplacian with nonlocal exterior conditions via forms method, J. Evol. Equ., 20 (2020), 1597-1631.  doi: 10.1007/s00028-020-00567-0.  Google Scholar

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J. DávilaM. Del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006.  Google Scholar

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R. GorenfloF. Mainardi and A. Vivoli, Continuous-time random walk and parametric subordination in fractional diffusion, Chaos, Solitons Fractals, 34 (2007), 87-103.  doi: 10.1016/j.chaos.2007.01.052.  Google Scholar

[23]

G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of $\mu$-transmission pseudodifferential operators, Adv. Math., 268 (2015), 478-528.  doi: 10.1016/j.aim.2014.09.018.  Google Scholar

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A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426.  Google Scholar

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V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, vol. 36, Masson, 1994.  Google Scholar

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T. KulczyckiM. KwaśnickiJ. Małecki and A. Stos, Spectral properties of the Cauchy process on half-line and interval, Proc. Lond. Math. Soc., 101 (2010), 589-622.  doi: 10.1112/plms/pdq010.  Google Scholar

[27]

M. Kwaśnicki, Eigenvalues of the fractional Laplace operator in the interval, J. Funct. Anal., 262 (2012), 2379-2402.   Google Scholar

[28]

N. Laskin, Fractional quantum mechanics, Phys. Rev. E, 62 (2000), 3135. Google Scholar

[29]

N. Laskin, Fractional schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

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N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Letters A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[31]

J. L. Lions, Contrôlabilité Exacte Perturbations et Stabilisation de Systèmes Distribués(Tome 1, Contrôlabilité Exacte. Tome 2, Perturbations), Recherches en mathematiques appliquées, Masson, 1988.  Google Scholar

[32]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar

[33]

J. L. Lions and E. Magenes, Problemes Aux Limites non Homogenes et Applications, Dunod, 1968. Google Scholar

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S. Longhi, Fractional Schrödinger equation in optics, Optics letters, 40 (2015), 1117-1120.   Google Scholar

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C. Louis-Rose and M. Warma, Approximate controllability from the exterior of space-time fractional wave equations, Appl. Math. Optim., 83 (2021), 207-250.  doi: 10.1007/s00245-018-9530-9.  Google Scholar

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M. M. Meerschaert, Fractional calculus, anomalous diffusion, and probability, in Fractional Dynamics: Recent Advances, World Scientific, 2012,265–284.  Google Scholar

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S. Micu and E. Zuazua, An introduction to the controllability of partial differential equations, in Quelques Questions de Théorie du Contrôle, Sari, T., ed., Collection Travaux en Cours Hermann, 2004. Google Scholar

[39]

S. I. Pohozaev, On the eigenfunctions of the equation $\delta u+ \lambda f(u) = 0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.   Google Scholar

[40]

J. Ralston, Gaussian beams and the propagation of singularities, Studies in Partial Differential Equations, 23 (1982), 206-248.   Google Scholar

[41]

J. RauchX. Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system, J. Math. Pures Appl., 84 (2005), 407-470.  doi: 10.1016/j.matpur.2004.09.006.  Google Scholar

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X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[43]

X. Ros-Oton and J. Serra, The extremal solution for the fractional Laplacian, Calc. Var. Partial Differential Equations, 50 (2014), 723-750.  doi: 10.1007/s00526-013-0653-1.  Google Scholar

[44]

X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Rat. Mech. Anal., 213 (2014), 587-628.  doi: 10.1007/s00205-014-0740-2.  Google Scholar

[45]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst, 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.  Google Scholar

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R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855.  doi: 10.1017/S0308210512001783.  Google Scholar

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R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Comm. Pure Appl. Anal., 12 (2013), 2445-2464.  doi: 10.3934/cpaa.2013.12.2445.  Google Scholar

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J. Simon, Compact sets in the space ${L}^p(0, {T}; {B})$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

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B. Stickler, Potential condensed-matter realization of space-fractional quantum mechanics: the one-dimensional Lévy crystal, Phys. Rev. E, 88 (2013), 012120. doi: 10.1103/PhysRevE.88.012120.  Google Scholar

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M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Springer Science & Business Media, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

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E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA, 49 (2009), 33-44.   Google Scholar

[52]

J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, in Nonlinear Partial Differential Equations, Springer, 7 (2012), 271–298. doi: 10.1007/978-3-642-25361-4_15.  Google Scholar

[53]

G. M. ViswanathanV. AfanasyevS. BuldyrevE. MurphyP. Prince and H. E. Stanley, Lévy flight search patterns of wandering albatrosses, Nature, 381 (1996), 413-415.  doi: 10.1038/381413a0.  Google Scholar

[54]

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

[55]

M. Warma, Approximate controllability from the exterior of space-time fractional diffusive equations, SIAM J. Control Optim., 57 (2019), 2037-2063.  doi: 10.1137/18M117145X.  Google Scholar

[56]

M. Warma and S. Zamorano, Null controllability from the exterior of a one-dimensional nonlocal heat equation, Control & Cybernetics, 48 (2019), 417-438.   Google Scholar

[57]

M. Warma and S. Zamorano, Analysis of the controllability from the exterior of strong damping nonlocal wave equations, ESAIM: Control Optim. Calc. Var., 26 (2020), 4Paper No. 42, 34 pp. doi: 10.1051/cocv/2019028.  Google Scholar

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show all references

References:
[1]

H. Antil, U. Biccari, R. Ponce, M. Warma and S. Zamorano, Controllability properties from the exterior under positivity constraints for a 1-d fractional heat equation, arXiv preprint, arXiv: 1910.14529. Google Scholar

[2]

H. Antil, R. Khatri and M. Warma, External optimal control of nonlocal PDEs, Inverse Problems, 35 (2019), 084003, 35 pp. doi: 10.1088/1361-6420/ab1299.  Google Scholar

[3]

C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.  Google Scholar

[4]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Berlin, 1976. Google Scholar

[5]

U. Biccari and V. Hernández-Santamarıa, Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects, IMA J. Math. Control Inf., 36 (2019), 1199-1235.  doi: 10.1093/imamci/dny025.  Google Scholar

[6]

U. Biccari and M. Warma, Null-controllability properties of a fractional wave equation with a memory term, Evol. Eq. Control. Theo., 9 (2020), 399-430.  doi: 10.3934/eect.2020011.  Google Scholar

[7]

U. BiccariM. Warma and E. Zuazua, Addendum: Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlin. Stud., 17 (2017), 837-839.  doi: 10.1515/ans-2017-6020.  Google Scholar

[8]

U. BiccariM. Warma and E. Zuazua, Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlin. Stud., 17 (2017), 387-409.  doi: 10.1515/ans-2017-0014.  Google Scholar

[9]

U. BiccariM. Warma and E. Zuazua, Controllability of the one-dimensional fractional heat equation under positivity constraints, Commun. Pure Appl. Anal., 19 (2020), 1949-1980.  doi: 10.3934/cpaa.2020086.  Google Scholar

[10]

M. Bologna, C. Tsallis and P. Grigolini, Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: exact time-dependent solutions, Phys. Rev. E, 62 (2000), 2213. doi: 10.1103/PhysRevE.62.2213.  Google Scholar

[11]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, vol. 13, Oxford University Press, 1998.  Google Scholar

[12]

B. Claus and M. Warma, Realization of the fractional Laplacian with nonlocal exterior conditions via forms method, J. Evol. Equ., 20 (2020), 1597-1631.  doi: 10.1007/s00028-020-00567-0.  Google Scholar

[13]

J.-M. Coron, Control and Nonlinearity, 136, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136.  Google Scholar

[14]

J. DávilaM. Del PinoS. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Analysis & PDE, 8 (2015), 1165-1235.  doi: 10.2140/apde.2015.8.1165.  Google Scholar

[15]

J. DávilaM. Del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006.  Google Scholar

[16]

E. Di NezzaG. 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

[17]

S. DipierroG. Palatucci and E. Valdinoci, Dislocation dynamics in crystals: A macroscopic theory in a fractional Laplace setting, Commun. Math. Phys., 333 (2015), 1061-1105.  doi: 10.1007/s00220-014-2118-6.  Google Scholar

[18]

A. A. DubkovB. Spagnolo and V. V. Uchaikin, Lévy flight superdiffusion: an introduction, Internat. J. Bifur. Chaos, 18 (2008), 2649-2672.  doi: 10.1142/S0218127408021877.  Google Scholar

[19]

A. FiscellaR. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math, 40 (2015), 235-253.  doi: 10.5186/aasfm.2015.4009.  Google Scholar

[20]

C. G. Gal and M. Warma, Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces, Comm. Partial Differential Equations, 42 (2017), 579-625.  doi: 10.1080/03605302.2017.1295060.  Google Scholar

[21]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.  doi: 10.1137/070698592.  Google Scholar

[22]

R. GorenfloF. Mainardi and A. Vivoli, Continuous-time random walk and parametric subordination in fractional diffusion, Chaos, Solitons Fractals, 34 (2007), 87-103.  doi: 10.1016/j.chaos.2007.01.052.  Google Scholar

[23]

G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of $\mu$-transmission pseudodifferential operators, Adv. Math., 268 (2015), 478-528.  doi: 10.1016/j.aim.2014.09.018.  Google Scholar

[24]

A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426.  Google Scholar

[25]

V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, vol. 36, Masson, 1994.  Google Scholar

[26]

T. KulczyckiM. KwaśnickiJ. Małecki and A. Stos, Spectral properties of the Cauchy process on half-line and interval, Proc. Lond. Math. Soc., 101 (2010), 589-622.  doi: 10.1112/plms/pdq010.  Google Scholar

[27]

M. Kwaśnicki, Eigenvalues of the fractional Laplace operator in the interval, J. Funct. Anal., 262 (2012), 2379-2402.   Google Scholar

[28]

N. Laskin, Fractional quantum mechanics, Phys. Rev. E, 62 (2000), 3135. Google Scholar

[29]

N. Laskin, Fractional schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[30]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Letters A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[31]

J. L. Lions, Contrôlabilité Exacte Perturbations et Stabilisation de Systèmes Distribués(Tome 1, Contrôlabilité Exacte. Tome 2, Perturbations), Recherches en mathematiques appliquées, Masson, 1988.  Google Scholar

[32]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar

[33]

J. L. Lions and E. Magenes, Problemes Aux Limites non Homogenes et Applications, Dunod, 1968. Google Scholar

[34]

S. Longhi, Fractional Schrödinger equation in optics, Optics letters, 40 (2015), 1117-1120.   Google Scholar

[35]

C. Louis-Rose and M. Warma, Approximate controllability from the exterior of space-time fractional wave equations, Appl. Math. Optim., 83 (2021), 207-250.  doi: 10.1007/s00245-018-9530-9.  Google Scholar

[36]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437.  doi: 10.1137/1010093.  Google Scholar

[37]

M. M. Meerschaert, Fractional calculus, anomalous diffusion, and probability, in Fractional Dynamics: Recent Advances, World Scientific, 2012,265–284.  Google Scholar

[38]

S. Micu and E. Zuazua, An introduction to the controllability of partial differential equations, in Quelques Questions de Théorie du Contrôle, Sari, T., ed., Collection Travaux en Cours Hermann, 2004. Google Scholar

[39]

S. I. Pohozaev, On the eigenfunctions of the equation $\delta u+ \lambda f(u) = 0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.   Google Scholar

[40]

J. Ralston, Gaussian beams and the propagation of singularities, Studies in Partial Differential Equations, 23 (1982), 206-248.   Google Scholar

[41]

J. RauchX. Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system, J. Math. Pures Appl., 84 (2005), 407-470.  doi: 10.1016/j.matpur.2004.09.006.  Google Scholar

[42]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[43]

X. Ros-Oton and J. Serra, The extremal solution for the fractional Laplacian, Calc. Var. Partial Differential Equations, 50 (2014), 723-750.  doi: 10.1007/s00526-013-0653-1.  Google Scholar

[44]

X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Rat. Mech. Anal., 213 (2014), 587-628.  doi: 10.1007/s00205-014-0740-2.  Google Scholar

[45]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst, 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.  Google Scholar

[46]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855.  doi: 10.1017/S0308210512001783.  Google Scholar

[47]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Comm. Pure Appl. Anal., 12 (2013), 2445-2464.  doi: 10.3934/cpaa.2013.12.2445.  Google Scholar

[48]

J. Simon, Compact sets in the space ${L}^p(0, {T}; {B})$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[49]

B. Stickler, Potential condensed-matter realization of space-fractional quantum mechanics: the one-dimensional Lévy crystal, Phys. Rev. E, 88 (2013), 012120. doi: 10.1103/PhysRevE.88.012120.  Google Scholar

[50]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Springer Science & Business Media, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[51]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA, 49 (2009), 33-44.   Google Scholar

[52]

J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, in Nonlinear Partial Differential Equations, Springer, 7 (2012), 271–298. doi: 10.1007/978-3-642-25361-4_15.  Google Scholar

[53]

G. M. ViswanathanV. AfanasyevS. BuldyrevE. MurphyP. Prince and H. E. Stanley, Lévy flight search patterns of wandering albatrosses, Nature, 381 (1996), 413-415.  doi: 10.1038/381413a0.  Google Scholar

[54]

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

[55]

M. Warma, Approximate controllability from the exterior of space-time fractional diffusive equations, SIAM J. Control Optim., 57 (2019), 2037-2063.  doi: 10.1137/18M117145X.  Google Scholar

[56]

M. Warma and S. Zamorano, Null controllability from the exterior of a one-dimensional nonlocal heat equation, Control & Cybernetics, 48 (2019), 417-438.   Google Scholar

[57]

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Figure 1.  The domain $ \Omega $ with the partition $ (\Gamma_0, \Gamma_1) $ of its boundary and the neighborhood $ \omega $ of $ \Gamma_0 $
Figure 2.  First 15 eigenvalues of the Dirichlet fractional Laplacian $ (-d_x^{\, 2})^{{s}}{} $ on $ (-1, 1) $ for $ s\in (0, 1/2] $ (left) and $ s\in(1/2, 1) $ (right)
Figure 3.  Gap between the first 15 eigenvalues of the Dirichlet fractional Laplacian $ (-d_x^{\, 2})^{{s}}{} $ on $ (-1, 1) $ for $ s\in(0, 1/2] $ (left) and $ s\in(1/2, 1) $ (right)
Figure 4.  Example of the domain $ \Omega $ with the partition of the boundary $ (\Gamma_0, \Gamma_1) $ and the two neighborhood of the boundary $ \widehat{\omega} $ and $ \omega $
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