# American Institute of Mathematical Sciences

July  2016, 36(7): 3719-3739. doi: 10.3934/dcds.2016.36.3719

## 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

Received  June 2015 Revised  December 2015 Published  March 2016

We study the interior approximate controllability of fractional wave equations with the fractional Caputo derivative associated with a non-negative self-adjoint operator satisfying the unique continuation property. Some well-posedness and fine regularity properties of solutions to fractional wave and fractional backward wave type equations are also obtained. As an example of applications of our results we obtain that if $1<\alpha<2$ and $\Omega\subset\mathbb{R}^N$ is a smooth connected open set with boundary $\partial\Omega$, then the system $\mathbb D_t^\alpha u+A_Bu=f$ in $\Omega\times (0,T)$, $u(\cdot,0)=u_0$, $\partial_tu(\cdot,0)=u_1$, is approximately controllable for any $T>0$, $(u_0,u_1)\in V_{\frac{1}{\alpha}}\times L^2(\Omega)$, $\omega\subset\Omega$ any open set and any $f\in C_0^\infty(\omega\times (0,T))$. Here, $A_B$ can be the realization in $L^2(\Omega)$ of a symmetric non-negative uniformly elliptic operator with Dirichlet or Robin boundary conditions, or the realization in $L^2(\Omega)$ of the fractional Laplace operator $(-\Delta)^s$ ($0< s <1$) with the Dirichlet boundary condition ($u=0$ on $\mathbb{R}^N\setminus\Omega$) and the space $V_{\frac{1}{\alpha}}$ denotes the domain of the fractional power of order $\frac{1}{\alpha}$ of the operator $A_B$.
Citation: Valentin Keyantuo, Mahamadi Warma. On the interior approximate controllability for fractional wave equations. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3719-3739. doi: 10.3934/dcds.2016.36.3719
##### 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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##### 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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