Non-Overlapping Domain Decomposition for 1D Optimal Control Problems Governed by Time-Fractional Diffusion Equations on Coupled Domains: Optimality System and Virtual Controls
Leugering G, Mehandiratta V, Mehra M (2024)
Publication Type: Journal article
Publication year: 2024
Journal
Book Volume: 8
Article Number: 129
Journal Issue: 3
DOI: 10.3390/fractalfract8030129
Abstract
We consider a non-overlapping domain decomposition method for optimal control problems of the tracking type governed by time-fractional diffusion equations in one space dimension, where the fractional time derivative is considered in the Caputo sense. We concentrate on a transmission problem defined on two adjacent intervals, where at the interface we introduce an iterative non-overlapping domain decomposition in the spirit of P.L. Lions for the corresponding first-order optimality system, such that the optimality system corresponding to the optimal control problem on the entire domain is iteratively decomposed into two systems on the respective sub-domains; this approach can be framed as first optimize, then decompose. We show that the iteration involving the states and adjoint states converges in the appropriate spaces. Moreover, we show that the decomposed systems on the sub-domain can in turn be interpreted as optimality systems of so-called virtual control problems on the sub-domains. Using this property, we are able to solve the original optimal control problem by an iterative solution of optimal control problems on the sub-domains. This approach can be framed as first decompose, then optimize. We provide a mathematical analysis of the problems as well as a numerical finite difference discretization using the L1-method with respect to the Caputo derivative, along with two examples in order to verify the method.
Authors with CRIS profile
Involved external institutions
Indian Institute of Technology (IIT), Delhi
India (IN)
King Abdullah University of Science and Technology (KAUST) / جامعة الملك عبد الله للعلوم و التقنية
Saudi Arabia (SA)
India (IN)
King Abdullah University of Science and Technology (KAUST) / جامعة الملك عبد الله للعلوم و التقنية
Saudi Arabia (SA)
How to cite
APA:
Leugering, G., Mehandiratta, V., & Mehra, M. (2024). Non-Overlapping Domain Decomposition for 1D Optimal Control Problems Governed by Time-Fractional Diffusion Equations on Coupled Domains: Optimality System and Virtual Controls. Fractal and Fractional, 8(3). https://doi.org/10.3390/fractalfract8030129
MLA:
Leugering, Günter, Vaibhav Mehandiratta, and Mani Mehra. "Non-Overlapping Domain Decomposition for 1D Optimal Control Problems Governed by Time-Fractional Diffusion Equations on Coupled Domains: Optimality System and Virtual Controls." Fractal and Fractional 8.3 (2024).
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