American Institute of Mathematical Sciences

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June  2020, 9(2): 399-430. doi: 10.3934/eect.2020011

Null-controllability properties of a fractional wave equation with a memory term

Umberto Biccari 1,, and  Mahamadi Warma 2,

1. 

DeustoTech, University of Deusto, 48007 Bilbao, Basque Country, Spain, Facultad de Ingeniería, Universidad de Deusto, Avenida de las Universidades 24, 48007 Bilbao, Basque Country, Spain
2. 

University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, Faculty of Natural Sciences, 17 University AVE. STE 1701 San Juan PR 00925-2537 (USA)

* Corresponding author: Umberto Biccari

Received  January 2019 Revised  May 2019 Published  June 2020 Early access  August 2019

Fund Project: The work of the first author is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement NO: 694126-DyCon), by the Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain), and by the ELKARTEK project KK-2018/00083 ROAD2DC of the Basque Government. The work of both authors is supported by the Air Force Office of Scientific Research (AFOSR) under Award NO: FA9550-18-1-0242

We study the null-controllability properties of a one-dimensional wave equation with memory associated with the fractional Laplace operator. The goal is not only to drive the displacement and the velocity to rest at some time-instant but also to require the memory term to vanish at the same time, ensuring that the whole process reaches the equilibrium. The problem being equivalent to a coupled nonlocal PDE-ODE system, in which the ODE component has zero velocity of propagation, we are required to use a moving control strategy. Assuming that the control is acting on an open subset $ \omega(t) $ which is moving with a constant velocity $ c\in\mathbb{R} $, the main result of the paper states that the equation is null controllable in a sufficiently large time $ T $ and for initial data belonging to suitable fractional order Sobolev spaces. The proof will use a careful analysis of the spectrum of the operator associated with the system and an application of a classical moment method.

Keywords: Fractional wave equation, memory, null controllability, moving control, moment method.
Mathematics Subject Classification: 30E05, 35L05, 35R11, 45K05, 93B05, 93B60, 93C05.
Citation: Umberto Biccari, Mahamadi Warma. Null-controllability properties of a fractional wave equation with a memory term. Evolution Equations and Control Theory, 2020, 9 (2) : 399-430. doi: 10.3934/eect.2020011
References:
[1]

U. Biccari, Internal control for non-local Schrödinger and wave equations involving the fractional Laplace operator, arXiv preprint, arXiv: 1411.7800.

[2]

U. Biccari and S. Micu, Null-controllability properties of the wave equation with a second order memory term, J. Differential Equations, 267 (2019), 1376-1422.  doi: 10.1016/j.jde.2019.02.009.

[3]

F. W. Chaves-Silva, L. Rosier and E. Zuazua, Null controllability of a system of viscoelasticity with a moving control, J. Math. Pures Appl. (9), 101 (2014), 198–222, https://doi.org/10.1016/j.matpur.2013.05.009. doi: 10.1016/j.matpur.2013.05.009.

[4]

F. W. Chaves-Silva, X. Zhang and E. Zuazua, Controllability of evolution equations with memory, SIAM J. Control Optim., 55 (2017), 2437–2459, https://doi.org/10.1137/151004239. doi: 10.1137/151004239.

[5]

B. Claus and M. Warma, Realization of the fractional laplacian with nonlocal exterior conditions via forms method, arXiv preprint, arXiv: 1904.13312.

[6]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573, https://doi.org/10.1016/j.bulsci.2011.12.004. doi: 10.1016/j.bulsci.2011.12.004.

[7]

A. A. Dubkov, B. Spagnolo and V. V. Uchaikin, Lévy flight superdiffusion: an introduction, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 2649–2672, https://doi.org/10.1142/S0218127408021877. doi: 10.1142/S0218127408021877.

[8]

C. G. Gal and M. Warma, Fractional in time semilinear parabolic equations and applications, HAL Id: hal-01578788, hhttps://hal.archives-ouvertes.fr/hal-01578788.

[9]

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, https://doi.org/10.1080/03605302.2017.1295060. doi: 10.1080/03605302.2017.1295060.

[10]

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

[11]

J.-P. Kahane, Pseudo-périodicité et séries de Fourier lacunaires, Ann. Sci. École Norm. Sup. (3), 79 (1962), 93–150, http://www.numdam.org/item?id=ASENS_1962_3_79_2_93_0. doi: 10.24033/asens.1108.

[12]

J. U. Kim, Control of a second-order integro-differential equation, SIAM J. Control Optim., 31 (1993), 101–110, https://doi.org/10.1137/0331008. doi: 10.1137/0331008.

[13]

M. Kwaśnicki, Eigenvalues of the fractional Laplace operator in the interval, J. Funct. Anal., 262 (2012), 2379–2402, https://doi.org/10.1016/j.jfa.2011.12.004. doi: 10.1016/j.jfa.2011.12.004.

[14]

G. Leugering, Exact controllability in viscoelasticity of fading memory type, Applicable Anal., 18 (1984), 221–243, https://doi.org/10.1080/00036818408839521. doi: 10.1080/00036818408839521.

[15]

G. Leugering, Exact boundary controllability of an integro-differential equation, Appl. Math. Optim., 15 (1987), 223–250, https://doi.org/10.1007/BF01442653. doi: 10.1007/BF01442653.

[16]

P. Loreti, L. Pandolfi and D. Sforza, Boundary controllability and observability of a viscoelastic string, SIAM J. Control Optim., 50 (2012), 820–844, https://doi.org/10.1137/110827740. doi: 10.1137/110827740.

[17]

P. Loreti and D. Sforza, Reachability problems for a class of integro-differential equations, J. Differential Equations, 248 (2010), 1711–1755, https://doi.org/10.1016/j.jde.2009.09.016. doi: 10.1016/j.jde.2009.09.016.

[18]

Q. Lü, X. Zhang and E. Zuazua, Null controllability for wave equations with memory, J. Math. Pures Appl. (9), 108 (2017), 500–531, https://doi.org/10.1016/j.matpur.2017.05.001. doi: 10.1016/j.matpur.2017.05.001.

[19]

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

[20]

P. Martin, L. Rosier and P. Rouchon, Null controllability of the structurally damped wave equation with moving control, SIAM J. Control Optim., 51 (2013), 660–684, https://doi.org/10.1137/110856150. doi: 10.1137/110856150.

[21]

M. I. Mustafa, On the control of the wave equation by memory-type boundary condition, Discrete Contin. Dyn. Syst., 35 (2015), 1179–1192, https://doi.org/10.3934/dcds.2015.35.1179. doi: 10.3934/dcds.2015.35.1179.

[22]

L. Pandolfi, Boundary controllability and source reconstruction in a viscoelastic string under external traction, J. Math. Anal. Appl., 407 (2013), 464–479, https://doi.org/10.1016/j.jmaa.2013.05.051. doi: 10.1016/j.jmaa.2013.05.051.

[23]

J. Prüss, Evolutionary Integral Equations and Applications, [2012] reprint of the 1993 edition, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1993, https://doi.org/10.1007/978-3-0348-8570-6. doi: 10.1007/978-3-0348-8570-6.

[24]

M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, vol. 35 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987.

[25]

I. Romanov and A. Shamaev, Exact controllability of the distributed system, governed by string equation with memory, J. Dyn. Control Syst., 19 (2013), 611–623, https://doi.org/10.1007/s10883-013-9199-y. doi: 10.1007/s10883-013-9199-y.

[26]

L. Rosier and B.-Y. Zhang, Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain, J. Differential Equations, 254 (2013), 141–178, https://doi.org/10.1016/j.jde.2012.08.014. doi: 10.1016/j.jde.2012.08.014.

[27]

W. R. Schneider, Grey noise, in Stochastic Processes, Physics and Geometry (Ascona and Locarno, 1988), World Sci. Publ., Teaneck, NJ, (1990), 676–681.

[28]

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.

[29]

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

[30]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009, https://doi.org/10.1007/978-3-7643-8994-9. doi: 10.1007/978-3-7643-8994-9.

[31]

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

[32]

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, https://doi.org/10.1007/s11118-014-9443-4. doi: 10.1007/s11118-014-9443-4.

[33]

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

[34]

M. Warma and S. Zamorano, Analysis of the controllability from the exterior of strong damping nonlocal wave equations, ESAIM: Control, Optimisation and Calculus of Variations (COCV), (2019), https://doi.org/10.1051/cocv/2019028. doi: 10.1051/cocv/2019028.

[35]

R. M. Young, An Introduction to Nonharmonic Fourier Series, 1st edition, Academic Press, Inc., San Diego, CA, 2001.

[36]

P. Zhuang and F. Liu, Implicit difference approximation for the time fractional diffusion equation, J. Appl. Math. Comput., 22 (2006), 87–99, https://doi.org/10.1007/BF02832039. doi: 10.1007/BF02832039.

show all references

References:
[1]

U. Biccari, Internal control for non-local Schrödinger and wave equations involving the fractional Laplace operator, arXiv preprint, arXiv: 1411.7800.

[2]

U. Biccari and S. Micu, Null-controllability properties of the wave equation with a second order memory term, J. Differential Equations, 267 (2019), 1376-1422.  doi: 10.1016/j.jde.2019.02.009.

[3]

F. W. Chaves-Silva, L. Rosier and E. Zuazua, Null controllability of a system of viscoelasticity with a moving control, J. Math. Pures Appl. (9), 101 (2014), 198–222, https://doi.org/10.1016/j.matpur.2013.05.009. doi: 10.1016/j.matpur.2013.05.009.

[4]

F. W. Chaves-Silva, X. Zhang and E. Zuazua, Controllability of evolution equations with memory, SIAM J. Control Optim., 55 (2017), 2437–2459, https://doi.org/10.1137/151004239. doi: 10.1137/151004239.

[5]

B. Claus and M. Warma, Realization of the fractional laplacian with nonlocal exterior conditions via forms method, arXiv preprint, arXiv: 1904.13312.

[6]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573, https://doi.org/10.1016/j.bulsci.2011.12.004. doi: 10.1016/j.bulsci.2011.12.004.

[7]

A. A. Dubkov, B. Spagnolo and V. V. Uchaikin, Lévy flight superdiffusion: an introduction, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 2649–2672, https://doi.org/10.1142/S0218127408021877. doi: 10.1142/S0218127408021877.

[8]

C. G. Gal and M. Warma, Fractional in time semilinear parabolic equations and applications, HAL Id: hal-01578788, hhttps://hal.archives-ouvertes.fr/hal-01578788.

[9]

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, https://doi.org/10.1080/03605302.2017.1295060. doi: 10.1080/03605302.2017.1295060.

[10]

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

[11]

J.-P. Kahane, Pseudo-périodicité et séries de Fourier lacunaires, Ann. Sci. École Norm. Sup. (3), 79 (1962), 93–150, http://www.numdam.org/item?id=ASENS_1962_3_79_2_93_0. doi: 10.24033/asens.1108.

[12]

J. U. Kim, Control of a second-order integro-differential equation, SIAM J. Control Optim., 31 (1993), 101–110, https://doi.org/10.1137/0331008. doi: 10.1137/0331008.

[13]

M. Kwaśnicki, Eigenvalues of the fractional Laplace operator in the interval, J. Funct. Anal., 262 (2012), 2379–2402, https://doi.org/10.1016/j.jfa.2011.12.004. doi: 10.1016/j.jfa.2011.12.004.

[14]

G. Leugering, Exact controllability in viscoelasticity of fading memory type, Applicable Anal., 18 (1984), 221–243, https://doi.org/10.1080/00036818408839521. doi: 10.1080/00036818408839521.

[15]

G. Leugering, Exact boundary controllability of an integro-differential equation, Appl. Math. Optim., 15 (1987), 223–250, https://doi.org/10.1007/BF01442653. doi: 10.1007/BF01442653.

[16]

P. Loreti, L. Pandolfi and D. Sforza, Boundary controllability and observability of a viscoelastic string, SIAM J. Control Optim., 50 (2012), 820–844, https://doi.org/10.1137/110827740. doi: 10.1137/110827740.

[17]

P. Loreti and D. Sforza, Reachability problems for a class of integro-differential equations, J. Differential Equations, 248 (2010), 1711–1755, https://doi.org/10.1016/j.jde.2009.09.016. doi: 10.1016/j.jde.2009.09.016.

[18]

Q. Lü, X. Zhang and E. Zuazua, Null controllability for wave equations with memory, J. Math. Pures Appl. (9), 108 (2017), 500–531, https://doi.org/10.1016/j.matpur.2017.05.001. doi: 10.1016/j.matpur.2017.05.001.

[19]

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

[20]

P. Martin, L. Rosier and P. Rouchon, Null controllability of the structurally damped wave equation with moving control, SIAM J. Control Optim., 51 (2013), 660–684, https://doi.org/10.1137/110856150. doi: 10.1137/110856150.

[21]

M. I. Mustafa, On the control of the wave equation by memory-type boundary condition, Discrete Contin. Dyn. Syst., 35 (2015), 1179–1192, https://doi.org/10.3934/dcds.2015.35.1179. doi: 10.3934/dcds.2015.35.1179.

[22]

L. Pandolfi, Boundary controllability and source reconstruction in a viscoelastic string under external traction, J. Math. Anal. Appl., 407 (2013), 464–479, https://doi.org/10.1016/j.jmaa.2013.05.051. doi: 10.1016/j.jmaa.2013.05.051.

[23]

J. Prüss, Evolutionary Integral Equations and Applications, [2012] reprint of the 1993 edition, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1993, https://doi.org/10.1007/978-3-0348-8570-6. doi: 10.1007/978-3-0348-8570-6.

[24]

M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, vol. 35 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987.

[25]

I. Romanov and A. Shamaev, Exact controllability of the distributed system, governed by string equation with memory, J. Dyn. Control Syst., 19 (2013), 611–623, https://doi.org/10.1007/s10883-013-9199-y. doi: 10.1007/s10883-013-9199-y.

[26]

L. Rosier and B.-Y. Zhang, Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain, J. Differential Equations, 254 (2013), 141–178, https://doi.org/10.1016/j.jde.2012.08.014. doi: 10.1016/j.jde.2012.08.014.

[27]

W. R. Schneider, Grey noise, in Stochastic Processes, Physics and Geometry (Ascona and Locarno, 1988), World Sci. Publ., Teaneck, NJ, (1990), 676–681.

[28]

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.

[29]

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

[30]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009, https://doi.org/10.1007/978-3-7643-8994-9. doi: 10.1007/978-3-7643-8994-9.

[31]

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

[32]

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, https://doi.org/10.1007/s11118-014-9443-4. doi: 10.1007/s11118-014-9443-4.

[33]

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

[34]

M. Warma and S. Zamorano, Analysis of the controllability from the exterior of strong damping nonlocal wave equations, ESAIM: Control, Optimisation and Calculus of Variations (COCV), (2019), https://doi.org/10.1051/cocv/2019028. doi: 10.1051/cocv/2019028.

[35]

R. M. Young, An Introduction to Nonharmonic Fourier Series, 1st edition, Academic Press, Inc., San Diego, CA, 2001.

[36]

P. Zhuang and F. Liu, Implicit difference approximation for the time fractional diffusion equation, J. Appl. Math. Comput., 22 (2006), 87–99, https://doi.org/10.1007/BF02832039. doi: 10.1007/BF02832039.

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