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

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 $$\mathbb D_t^{\mu,\nu} u+A_Bu=f|_{\omega}\;\; \mbox{ in }\; \Omega\times (0,T),\,\, (\mathbb I_t^{(1-\nu)(1-\mu)}u)(\cdot,0)=u_0 \mbox{ in }\;\Omega,$$ 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, $\mathcal N^su+\beta u=0$ in $\mathbb R^N\setminus\overline{\Omega}$.