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 AB with compact resolvent on L2(Ω), where ${\Omega }\subset \mathbb {R}^{N}$ (N ≥ 1) is a bounded open set. More precisely, we show that if 0 ≤ ν ≤ 1 and 0 < μ ≤ 1, then the system $$ \mathbb{D}_{t}^{\mu,\nu} u+A_{B}u=f\chi_{\omega}\quad\text{ in }~{\Omega}\times (0,T),\qquad (\mathbb{I}_{t}^{(1-\nu)(1-\mu)}u)(\cdot,0)=u_{0}\quad \text{ in }~{\Omega}, $$ is approximately controllable in any time T > 0, u0 ∈ L2(Ω) and any nonempty open set ω ⊂Ω and χω is the characteristic function of ω. In addition, if the operator AB has the unique continuation property, then the system is also mean (memory) approximately controllable. The operator AB can be the realization in L2(Ω) of a symmetric, non-negative uniformly elliptic second order operator with Dirichlet or Robin boundary conditions, or the realization in L2(Ω) of the fractional Laplace operator (−Δ)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 $\mathbb {R}^{N}\setminus \overline {\Omega }$ .