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 A_B 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 χ_ω is the characteristic function of ω. In addition, if the operator A_B has the unique continuation property, then the system is also mean (memory) approximately controllable. The operator A_B 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 $\mathbb {R}^{N}\setminus \overline {\Omega }$.