Well‐posedness, optimal control and discretization for time‐fractional parabolic equations with time‐dependent coefficients on metric graphs. Issue 3 (18th October 2022)

• Record Type: Journal Article
• Authors: Mehandiratta, Vaibhav; Mehra, Mani; Leugering, Günter
• Is Part Of: Asian journal of control. Volume 25:Issue 3(2023)
• Page Start: 2360
• Page End: 2377
• Publication Date: 2022-10-18
• Subjects: Caputo fractional derivative -- distributed control -- metric graph -- time‐fractional parabolic equations; Automatic control -- Periodicals; Control theory -- Periodicals; 629.805
• DOI: 10.1002/asjc.2958
• Languages: English
• ISSNs: 1561-8625

Abstract

Abstract: We study distributed optimal control problems governed by time‐fractional parabolic equations with time dependent coefficients on metric graphs, where the fractional derivative is considered in the Caputo sense. Using the Galerkin method and compactness results, for the spatial part, and approximating the kernel of the time‐fractional Caputo derivative by a sequence of more regular kernel functions, we first prove the well‐posedness of the system. We then turn to the existence and uniqueness of solutions to the distributed optimal control problem. By means of the Lagrange multiplier method, we develop an adjoint calculus for the right Caputo derivative and derive the corresponding first order optimality system. Moreover, we propose a finite difference scheme to find the approximate solution of the state equation and the resulting optimality system on metric graphs. Finally, examples are provided on two different graphs to illustrate the performance of the proposed difference scheme.