A Fractional Space-time Optimal Control Problem: Analysis and Discretization∗

We study a linear-quadratic optimal control problem involving a parabolic equation with fractional diffusion and Caputo fractional time derivative of orders s ∈ (0, 1) and γ ∈ (0, 1], respectively. The spatial fractional diffusion is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic operator. Thus, we consider an equivalent formulation with a quasi-stationary elliptic problem with a dynamic boundary condition as state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We consider a fully-discrete scheme: piecewise constant functions for the control and, for the state, firstdegree tensor product finite elements in space and a finite difference discretization in time. We show convergence of this scheme and, for s ∈ (0, 1) and γ = 1, we derive a priori error estimates.