Approximate Controllability from the Exterior of Space-Time Fractional Wave Equations - Louis-Rose, Carole; Warma, Mahamadi (Applied mathematics & optimization, 2021)
We investigate the controllability from the exterior of space-time fractional wave equations involving the Caputo time fractional derivative with the fractional Laplace operator subject to nonhomogeneous Dirichlet or Robin type exterior conditions. We prove that if 1 < α < 2 , 0 < s < 1 and Ω ⊂ R N is a bounded Lipschitz domain, then the system D t α u + ( - Δ ) s u = 0 in Ω × ( 0 , T ) , B u = g in ( R N \ Ω ) × ( 0 , T ) , u ( · , 0 ) = u 0 , ∂ t u ( · , 0 ) = u 1 in Ω , is approximately controllable for any T > 0 , ( u 0 , u 1 ) ∈ L 2 ( Ω ) × V B - 1 α and every g ∈ D ( O × ( 0 , T ) ) , where O ⊂ ( R N \ Ω ) is any non-empty open set in the case of the Dirichlet exterior condition B u = u , and O ⊆ R N \ Ω is any open set dense in R N \ Ω for the Robin exterior conditions B u : = N s u + κ u . Here, N s u is the nonlocal normal derivative of u and V B - 1 α denotes the dual of the domain of the fractional power of order 1 α of the realization in L 2 ( Ω ) of the operator ( - Δ ) s with the zero (Dirichlet or Robin) exterior conditions B u = 0 in R N \ Ω .