Approximate controllabilty from the exterior of space-time fractional diffusive equations

Abstract

Let $\Om\subset\RR^N$ a bounded domain with a Lipschitz continuous boundary. We study the controllability of the space-time fractional diffusion equation \begin{equation*} \begin{cases} \mathbb D_t^\alpha u+(-\Delta)^su=0\;\;&\mbox{ in }\;(0,T)\times\Omega\\ u=g &\mbox{ in }\;(0,T)\times(\RR^N\setminus\Omega)\\ u(0,\cdot)=u_0&\mbox{ in }\;\Omega, \end{cases} \end{equation*} where $u=u(t,x)$ is the state to be controlled and $g=g(t,x)$ is the control function which is localized in a subset $\mathcal O$ of $\Omc$. Here, $0<\alpha\le 1$, $0<s<1$ and $T>0$ be real numbers. After giving an explicit representation of solutions, we show that the system is always approximately controllable for every $T>0$, $u_0\in L^2(\Omega)$ and $g\in \mathcal D((0,T)\times\mathcal O)$ where $\mathcal O\subset(\RR^N\setminus\bOm)$ is any open set. The results obtained are sharp in the sense that such a system is never null controllable if $0<\alpha<1$. The proof of our result is based on a new unique continuation principle for the eigenvalues problem associated with the fractional Laplace operator subject to the zero exterior boundary condition that we have established.