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 \(0\le\nu\le 1\), \(00\), \(u_0\in L^2(\Omega)\) 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