An Inverse Problem of Identifying Two Coefficients in a Time-Fractional Reaction Diffusion System

In this paper, we aim to study an inverse problem for determining two time-independent coefficients in a fractional diffusion system from the final measurements. First, we prove the well-posedness of the state problem, and then we show some regularity results for the solution of the direct system using the Mittag-Leffler function. Then, we reformulate our inverse problem into an optimal control one. Afterwards, we establish the existence of the minimizer and prove the stability estimate for two coefficients with respect to the final data. The descent method is proposed as a numerical one based on the gradient calculus via the adjoint state and we compare it with the conjugate gradient method. Finally, we will present some numerical tests that shows the efficiency of the proposed methods.