Optimal Control of a Fractional Diffusion Equation with Delay

We study a homogeneous Dirichlet boundary fractional diffusion equation with delay in a bounded domain. The fractional time derivative is considered in the left Caputo sense. By means of a linear continuous operator, we first transform the fractional diffusion equation with delay into a an equivalent equation without delay. Then we show that the optimal control problem associate to the controlled equivalent fractional diffusion equation has a unique solution. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Caputo derivative, we obtain an optimality system