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 $L1$ 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.


  1. time-fractional diffusion equation
  2. Caputo fractional derivative
  3. metric graph
  4. optimal control

MSC codes

  1. 35R11
  2. 49J20
  3. 26A33
  4. 49K20
  5. 93C20

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Information & Authors


Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 4216 - 4242
ISSN (online): 1095-7138


Submitted: 26 May 2020
Accepted: 3 July 2021
Published online: 4 November 2021


  1. time-fractional diffusion equation
  2. Caputo fractional derivative
  3. metric graph
  4. optimal control

MSC codes

  1. 35R11
  2. 49J20
  3. 26A33
  4. 49K20
  5. 93C20



Funding Information

Deutscher Akademischer Austauschdienst https://doi.org/10.13039/501100001655 : 1-3/2016
Medizinische Fakultät, Friedrich-Alexander-Universität Erlangen-Nürnberg https://doi.org/10.13039/501100009508
University Grants Commission https://doi.org/10.13039/501100001501 : 1-3/2016

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