Memory approximate controllability properties for higher order Hilfer time fractional evolution equations

In this paper we study the approximate controllability of fractional partial differential equations associated with the so-called Hilfer type time fractional derivative and a non-negative selfadjoint operator $A$ with a compact resolvent on $L^2(\mathrm{\Omega })$, where $\mathrm{\Omega }\subset {\text{RR}}^N$ ($N\ge 1$) is an open set. More precisely, we show that if $0\le \nu \le 1$, $1<\mu \le 2$ and $\mathrm{\Omega }\subset {\text{RR}}^N$ is an open set, then the system \begin{equation*} \begin{cases} \D^{μ,ν}_tu+Au=fχ_ω;;&\mbox{ in };Ω\times(0,T),\ (I_t^{(1-ν)(2-μ)}u)(\cdot,0)=u_0 &\mbox{ in };Ω,\ (\partial_tI_t^{(1-ν)(2-μ)}u)(\cdot,0)=u_1 &\mbox{ in };Ω, \end{cases} \end{equation*} is memory approximately controllable for any $T>0$, $u_0\in D(A^{1\mathrm{/}\mu })$, $u_1\in L^2(\mathrm{\Omega })$ and any non-empty open set $\omega \subset \mathrm{\Omega }$. The same result holds for every $u_0\in D(A^{1\mathrm{/}2})$ and $u_1\in L^2(\mathrm{\Omega })$.