Discretization and Analysis of an Optimal Control of a Variable-Order Time-Fractional Diffusion Equation with Pointwise Constraints

We prove the well-posedness and regularity of an optimal control model with pointwise constraints governed by a variable-order Caputo time-fractional diffusion equation (tFDE), in which the adjoint equation reduces to a Riemann–Liouville tFDE with a different type of variable-order fractional differential operator. We develop and analyze a finite element discretization to the optimal control model without any regularity assumptions of the true solution. Numerical experiments are performed to substantiate the theoretical findings.