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

Abstract

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 $L^2(\Omega )$, where $Ω\subset\RR^N$ ($N\ge 1$) is an open set. More precisely, we show that if $0\le \nu \le 1$, $1<\mu \le 2$ and $Ω\subset\RR^N$ 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 $T>0$, $u_0\in D(A^{1/\mu })$, $u_1\in L^2(\Omega )$ and any non-empty open set $\omega \subset \Omega$. The same result holds for every $u_0\in D(A^{1/2})$ and $u_1\in L^2(\Omega )$.

arXiv admin note: text overlap with arXiv:2003.08188