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On the interior approximate controllability for fractional wave equations
1. | University of Puerto Rico, Río Piedras Campus,, Department of Mathematics, P.O. Box 70377, San Juan PR 00936-8377, United States |
2. | University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, P.O. Box 70377, San Juan PR 00936-8377 |
References:
[1] |
O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys., 40 (2007), 6287-6303.
doi: 10.1088/1751-8113/40/24/003. |
[2] |
R. Almeida and D. Torres, Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1490-1500.
doi: 10.1016/j.cnsns.2010.07.016. |
[3] |
E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001. |
[4] |
U. Biccari, Internal control for non-local Schrödinger and wave equations involving the fractional Laplace operator, arXiv:1411.7800. |
[5] |
M. M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397.
doi: 10.1080/03605302.2013.825918. |
[6] |
K. Fujishiro, Approximate controllability for fractional diffusion equations by Dirichlet boundary conditions, arXiv:1404.0207. |
[7] |
K. Fujishiro and M. Yamamoto, Approximate controllability for fractional diffusion equations by interior control, Appl. Anal., 93 (2014), 1793-1810.
doi: 10.1080/00036811.2013.850492. |
[8] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. |
[9] |
R. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, A. Carpinteri and F. Mainardi (Editors): Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Wien and New York, 378 (1997), 223-276. |
[10] |
R. Gorenflo and F. Mainardi, On Mittag-Leffler-type functions in fractional evolution processes, J. Comp. Appl. Math., 118 (2000), 283-299.
doi: 10.1016/S0377-0427(00)00294-6. |
[11] |
V. Keyantuo, C. Lizama and M. Warma, Existence, regularity and representation of solutions of time fractional diffusion equations, Adv. Differential Equations, to appear. |
[12] |
V. Keyantuo, C. Lizama and M. Warma, Existence, regularity and representation of solutions of fractional wave equations, Submitted. |
[13] |
Q. Lü and E. Zuazua, On the lack of controllability of fractional in time ODE and PDE, Mathematics of Control, Signals, and Systems, to appear. |
[14] |
F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, In Fractals and Fractional Calculus in Continuum Mechanics (Eds. A. Carpinteri and F. Mainardi), Springer Verlag, Wien, 378 (1997), 291-348.
doi: 10.1007/978-3-7091-2664-6_7. |
[15] |
K. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, New York: John Wiley & Sons Inc., 1993. |
[16] |
I. Podlubny, Fractional Differential Equations, 198 Academic Press, San Diego, California, USA, 1999. |
[17] |
K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.
doi: 10.1016/j.jmaa.2011.04.058. |
[18] |
R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. |
[19] |
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. |
[20] |
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. |
[21] |
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. |
[22] |
M. Warma, The fractional Neumann and Robin boundary condition for the fractional $p$-Laplacian on open sets, NoDEA Nonlinear Differential Equations Appl., 23 (2016), p1.
doi: 10.1007/s00030-016-0354-5. |
[23] |
E. Zuazua, Controllability of Partial Differential Equations, 3ème cycle. Castro Urdiales, Espagne, 2006. |
show all references
References:
[1] |
O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys., 40 (2007), 6287-6303.
doi: 10.1088/1751-8113/40/24/003. |
[2] |
R. Almeida and D. Torres, Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1490-1500.
doi: 10.1016/j.cnsns.2010.07.016. |
[3] |
E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001. |
[4] |
U. Biccari, Internal control for non-local Schrödinger and wave equations involving the fractional Laplace operator, arXiv:1411.7800. |
[5] |
M. M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397.
doi: 10.1080/03605302.2013.825918. |
[6] |
K. Fujishiro, Approximate controllability for fractional diffusion equations by Dirichlet boundary conditions, arXiv:1404.0207. |
[7] |
K. Fujishiro and M. Yamamoto, Approximate controllability for fractional diffusion equations by interior control, Appl. Anal., 93 (2014), 1793-1810.
doi: 10.1080/00036811.2013.850492. |
[8] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. |
[9] |
R. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, A. Carpinteri and F. Mainardi (Editors): Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Wien and New York, 378 (1997), 223-276. |
[10] |
R. Gorenflo and F. Mainardi, On Mittag-Leffler-type functions in fractional evolution processes, J. Comp. Appl. Math., 118 (2000), 283-299.
doi: 10.1016/S0377-0427(00)00294-6. |
[11] |
V. Keyantuo, C. Lizama and M. Warma, Existence, regularity and representation of solutions of time fractional diffusion equations, Adv. Differential Equations, to appear. |
[12] |
V. Keyantuo, C. Lizama and M. Warma, Existence, regularity and representation of solutions of fractional wave equations, Submitted. |
[13] |
Q. Lü and E. Zuazua, On the lack of controllability of fractional in time ODE and PDE, Mathematics of Control, Signals, and Systems, to appear. |
[14] |
F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, In Fractals and Fractional Calculus in Continuum Mechanics (Eds. A. Carpinteri and F. Mainardi), Springer Verlag, Wien, 378 (1997), 291-348.
doi: 10.1007/978-3-7091-2664-6_7. |
[15] |
K. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, New York: John Wiley & Sons Inc., 1993. |
[16] |
I. Podlubny, Fractional Differential Equations, 198 Academic Press, San Diego, California, USA, 1999. |
[17] |
K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.
doi: 10.1016/j.jmaa.2011.04.058. |
[18] |
R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. |
[19] |
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. |
[20] |
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. |
[21] |
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. |
[22] |
M. Warma, The fractional Neumann and Robin boundary condition for the fractional $p$-Laplacian on open sets, NoDEA Nonlinear Differential Equations Appl., 23 (2016), p1.
doi: 10.1007/s00030-016-0354-5. |
[23] |
E. Zuazua, Controllability of Partial Differential Equations, 3ème cycle. Castro Urdiales, Espagne, 2006. |
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