Approximate and mean approximate controllability properties for Hilfer time-fractional differential equations

Abstract

We study the approximate and mean approximate controllability properties of fractional partial differential equations associated with the so-called Hilfer type time-fractional derivative and a non-negative selfadjoint operator $A_B$ with a compact resolvent on $L^2(\Omega )$, where $\Omega \subset {\mathbb{R}}^N$ ($N\ge 1$) is a bounded open set. More precisely, we show that if $0\le \nu \le 1$, $0<\mu \le 1$ and $\Omega \subset {\mathbb{R}}^N$ is a bounded open set, then the system $$\mathbb D_t^{μ,ν} u+A_Bu=f|_ω\;\; \mbox{ in }\; Ω\times (0,T),\,\, (\mathbb I_t^{(1-ν)(1-μ)}u)(\cdot,0)=u_0 \mbox{ in }\;Ω,$$ is approximately controllable for any $T>0$, $u_0\in L^2(\Omega )$ and any non-empty open set $\omega \subset \Omega$. In addition, if the operator $A_B$ has the unique continuation property, then the system is also mean approximately controllable. The operator $A_B$ can be the realization in $L^2(\Omega )$ of a symmetric, non-negative uniformly elliptic second order 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 exterior condition, $u=0$ in ${\mathbb{R}}^N\setminus \Omega$, or the nonlocal Robin exterior condition, ${\mathcal{N}}^{s}u+\beta u=0$ in ${\mathbb{R}}^N\setminus \overline{\Omega }$.