Approximate and mean approximate controllability properties for Hilfer time-fractional differential equations

2020/03/18 by Aragones, Ernest, Keyantuo, Valentin, Warma, Mahamadi
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2003.08188

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 a compact resolvent on L²(Ω), where Ω⊂ℝ^N (N≥ 1) is a bounded open set. More precisely, we show that if 0≤ν≤ 1, 00, u₀∈ L²(Ω) and any non-empty open set ω⊂Ω. 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 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

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