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

Abstract

We investigate the controllability from the exterior of space-time fractional wave equations involving the Caputo time fractional derivative with the fractional Laplace operator subject to nonhomogeneous Dirichlet or Robin type exterior conditions. We prove that if $1<\alpha < 2$, $0<s<1$ and $\Omega \subset {{\mathbb {R}}}^N$ is a bounded Lipschitz domain, then the system

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathbb {D}}_t^\alpha u+(-\Delta )^su=0\;\;&{} \text{ in } \;\;\Omega \times (0,T), \\ Bu=g\;&{} \text{ in } \; ({\mathbb {R}}^N\setminus \Omega )\times (0,T), \\ u(\cdot ,0)=u_0, \;\;\partial _tu(\cdot ,0)=u_1\;&{} \text{ in } \;\Omega , \end{array}\right. } \end{aligned}$$

is approximately controllable for any $T>0$, $(u_0,u_1)\in L^2(\Omega )\times {\mathbb {V}}_B^{-\frac{1}{\alpha }}$ and every $g\in {\mathcal {D}}({\mathcal {O}}\times (0,T))$, where ${\mathcal {O}}\subset ({\mathbb {R}}^N\setminus \Omega )$ is any non-empty open set in the case of the Dirichlet exterior condition $Bu=u$, and ${\mathcal {O}}\subseteq {\mathbb {R}}^N\setminus \Omega$ is any open set dense in ${\mathbb {R}}^N\setminus \Omega$ for the Robin exterior conditions $Bu:=\mathcal N_su+\kappa u$. Here, ${\mathcal {N}}_su$ is the nonlocal normal derivative of u and ${\mathbb {V}}_B^{-\frac{1}{\alpha }}$ denotes the dual of the domain of the fractional power of order $\frac{1}{\alpha }$ of the realization in $L^2(\Omega )$ of the operator $(-\Delta )^s$ with the zero (Dirichlet or Robin) exterior conditions $Bu=0$ in ${{\mathbb {R}}}^N\setminus \Omega$.