Approximate Controllability from the Exterior of Space-Time Fractional Diffusive Equations

Abstract

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain with a Lipschitz continuous boundary. We study the controllability of the space-time fractional diffusive equation $\{\mathbb D_t^\alpha u+(-\Delta)^su=0 \mbox{ in}\;(0,T)\times\Omega, u=g\chi_{(0,T)\times\mathcal O} %& \mbox{ in}\;(0,T)\times(\mathbb{R}^N\setminus\Omega), u(0,\cdot)=u_0%& \mbox{ in}\;\Omega\}$, 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 nonempty open subset $\mathcal O$ of $\mathbb{R}^N\setminus\Omega$. Here, $0<\alpha\le 1$, $0<s<1,$ and $T>0$ are 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 $\mathcal O\subseteq\mathbb{R}^N\setminus\Omega$ is an arbitrary nonempty 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 Dirichlet exterior condition that we have established.