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Stochastic porous media equations with divergence Itô noise
Null-controllability properties of a fractional wave equation with a memory term
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) |
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.
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[1] |
U. Biccari, Internal control for non-local Schrödinger and wave equations involving the fractional Laplace operator, arXiv preprint, arXiv: 1411.7800. |
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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. |
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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.
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doi: 10.1137/151004239. |
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B. Claus and M. Warma, Realization of the fractional laplacian with nonlocal exterior conditions via forms method, arXiv preprint, arXiv: 1904.13312. |
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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.
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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.
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[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.
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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.
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[14] |
G. Leugering, Exact controllability in viscoelasticity of fading memory type, Applicable Anal., 18 (1984), 221–243, https://doi.org/10.1080/00036818408839521.
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[15] |
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[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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