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<p>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.</p>
</abstract>

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.

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

Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)

Control and Cybernetics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

We study distributed optimal control problems, governed by space-time fractional parabolic equations (STFPEs) involving time-fractional Caputo derivatives and...

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

Published Online: Aug 12, 2022

Page range: 191 - 226

Received: Feb 01, 2022

Accepted: May 01, 2022

DOI: https://doi.org/10.2478/candc-2022-0014

Keywords
space-time fractional parabolic equations, Caputo fractional derivative, distributed control, L1 method, Grünwald-Letnikov formula

© 2022 Vaibhav Mehandiratta et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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

Published Online: Aug 12, 2022

Page range: 191 - 226

Received: Feb 01, 2022

Accepted: May 01, 2022

DOI: https://doi.org/10.2478/candc-2022-0014

Keywordsspace-time fractional parabolic equations, Caputo fractional derivative, distributed control, L1 method, Grünwald-Letnikov formula

© 2022 Vaibhav Mehandiratta et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
space-time fractional parabolic equations, Caputo fractional derivative, distributed control, L1 method, Grünwald-Letnikov formula