Optimal Control for Fractional Diffusion Equations with Incomplete Data

We are concerned with the optimal control of time-fractional diffusion equations with missing boundary condition. Using the notion of no-regret control and least (or low) regret control developed by Lions, we first prove that the least regret control problem associated with the boundary fractional diffusion equation has a unique solution. Then we show that this solution converges to the no-regret control which we characterize by a singular optimality system.