Optimal control problems governed by space-time fractional parabolic equations on metric graphs

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Abstract
In this article, we study optimal control problems driven by space-time fractional parabolic equations (STFPEs) of Sturm-Liouville type on metric graphs, considering both distributed and boundary controls. The considered problem involves the fractional time derivative in the Caputo sense, and the spatial fractional derivative is of the Sturm-Liouville type which is the composition of the right-sided Caputo fractional derivative and left-sided Riemann-Liouville fractional derivative. By introducing the appropriate function spaces for the involved fractional operators in both the time and spatial variables, we first establish well-posedness results for the weak solution for the governing STFPEs using the Galerkin approximation method. Then, we prove that the associated optimal control problem possesses a unique optimal solution. Moreover, to characterize the control variables, we develop an adjoint problem involving a right-sided Caputo fractional derivative in time and derive optimality conditions for the considered problem, using the Lagrange multiplier method. Finally, we propose a difference scheme to find the numerical solution of the considered optimal control problem, particularly on a metric star graph by approximating the Caputo time derivative using the L1 method and spatial fractional derivative with the Grünwald-Letnikov formula. The performance and the accuracy of the proposed difference scheme have been illustrated via an example.

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