OPTIMAL CONTROL PROBLEMS DRIVEN BY TIME-FRACTIONAL DIFFUSION EQUATIONS ON METRIC GRAPHS: OPTIMALITY SYSTEM AND FINITE DIFFERENCE APPROXIMATION

We study optimal control problems for time-fractional diffusion equations on metric graphs, where the fractional derivative is considered in the Caputo sense. Using eigenfunction expansions for the spatial part, we first prove the well-posedness of the system. We then prove the existence of a unique solution to the optimal control problem, where we admit both boundary and distributed controls. We develop an adjoint calculus for the right Caputo derivative and derive the corresponding first order optimality system. We also propose a finite difference approximation to find the numerical solution of the optimality system on the graph. In the proposed method, the so-called Ll method is used for the discrete approximation of the Caputo derivative, while the space derivative is approximated using a standard central difference scheme, which results in converting the optimality system into a system of algebraic equations. Finally, an example is provided to demonstrate the performance of the numerical method.

Automated summary

The study investigates optimal control problems for time-fractional diffusion equations on metric graphs using Caputo fractional derivative. It proves the well-posedness of the system with eigenfunction expansions and establishes the existence of a unique solution to the optimal control problem. An adjoint calculus for the right Caputo derivative is developed, leading to a first-order optimality system. A finite difference approximation is proposed to find the numerical solution on the graph, with Ll method used for discrete approximation of Caputo derivative and standard central difference scheme for space derivative. An example is provided to demonstrate the performance of the numerical method.