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

TL;DR

This paper investigates the approximate controllability of space-time fractional diffusion equations with exterior controls, establishing that such systems are always approximately controllable but never null controllable when the fractional time derivative order is less than one.

Contribution

The paper introduces a new unique continuation principle for fractional Laplace eigenvalue problems and proves approximate controllability for fractional diffusion equations with exterior controls.

Findings

System is always approximately controllable for any positive time.

System is never null controllable if the fractional time derivative order is less than one.

Established a new unique continuation principle for fractional Laplace eigenvalue problems.

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.