Distributed optimal control problems driven by space-time fractional parabolic equations

We study distributed optimal control problems, governed by space-time fractional parabolic equations (STFPEs) involving time-fractional Caputo derivatives and spatial fractional derivatives of Sturm-Liouville type. We first prove existence and uniqueness of solutions of STFPEs on an open bounded interval and study their regularity. Then we show existence and uniqueness of solutions to a quadratic distributed optimal control problem. We derive an adjoint problem using the right-Caputo derivative in time and provide optimality conditions for the control problem. Moreover, we propose a finite difference scheme to find the approximate solution of the considered optimal control problem. In the proposed scheme, the well-known L1 method has been used to approximate the time-fractional Caputo derivative, while the spatial derivative is approximated using the Grünwald-Letnikov formula. Finally, we demonstrate the accuracy and the performance of the proposed difference scheme via examples.