On the interior approximate controllability for fractional wave equations

Abstract

We study the interior approximate controllability of fractional wave equations with the fractional Caputo derivative associated with a nonnegative self-adjoint operator satisfying the unique continuation property. Some well-posedness and fine regularity properties of solutions to fractional wave and fractional backward wave type equations are also obtained. As an example of applications of our results we obtain that if 1 < α < 2 and Ω⊂ ℝN is a smooth connected open set with boundary ∂Ω, then the system Dtα u + ABu = f in Ω × (0, T), u(·, 0) = u0, ∂tu(·, 0) = u1, is approximately controllable for any T > 0, (u0, u1) ∈ V 1/α×L2(Ω), ω ⊂ Ω any open set and any f ∈ C0∞ (Ω×(0, T)). Here, AB can be the realization in L2(Ω) of a symmetric non-negative uniformly elliptic 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 boundary condition (u = 0 on ℝN\Ω ) and the space V1/α denotes the domain of the fractional power of order 1/α of the operator AB.