On the interior approximate controllability for fractional wave equations

Valentin  Keyantuo; Mahamadi Warma

doi:10.3934/dcds.2016.36.3719

Article Contents

2016, Volume 36, Issue 7: 3719-3739. Doi: 10.3934/dcds.2016.36.3719

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On the interior approximate controllability for fractional wave equations

Valentin Keyantuo^1, and
Mahamadi Warma^2,

1.
University of Puerto Rico, Río Piedras Campus,, Department of Mathematics, P.O. Box 70377, San Juan PR 00936-8377
2.
University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, P.O. Box 70377, San Juan PR 00936-8377

Received: May 31, 2015

Revised: November 30, 2015

Published: May 2017

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Abstract

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 $A_B$.

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Mathematics Subject Classification: 93B05, 26A33, 35R11.

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