February  2022, 11(1): 301-324. 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  February 2022 Early access  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 and Control Theory, 2022, 11 (1) : 301-324. doi: 10.3934/eect.2021014
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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[24]

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

[25]

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

[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.

[27]

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

[28]

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

[29]

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

[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.

[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.

[32]

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

[33]

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

[34]

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

[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.

[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.

[37]

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

[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.

[39]

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

[40]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[51]

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

[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.

[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.

[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.

[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.

[56]

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

[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.

[58]

G. B. Whitham, Linear and Nonlinear Waves, vol. 42, John Wiley & Sons, 1999. doi: 10.1002/9781118032954.

[59]

K. Yosida, Functional Analysis, vol. 6, Springer-Verlag, Berlin New York, 1980.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[24]

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

[25]

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

[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.

[27]

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

[28]

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

[29]

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

[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.

[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.

[32]

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

[33]

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

[34]

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

[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.

[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.

[37]

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

[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.

[39]

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

[40]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[51]

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

[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.

[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.

[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.

[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.

[56]

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

[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.

[58]

G. B. Whitham, Linear and Nonlinear Waves, vol. 42, John Wiley & Sons, 1999. doi: 10.1002/9781118032954.

[59]

K. Yosida, Functional Analysis, vol. 6, Springer-Verlag, Berlin New York, 1980.

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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