Optimal control problems of parabolic fractional Sturm-Liouville equations in a star graph

Abstract

<p style='text-indent:20px;'>In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm–Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler–Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary controls. The existence and uniqueness of minimizers, and the characterization of the first order optimality conditions are obtained in a general star graph by using the method of Lagrange multipliers.</p>

Non-Overlapping Domain Decomposition for 1D Optimal Control Problems Governed by Time-Fractional Diffusion Equations on Coupled Domains: Optimality System and Virtual Controls

“…We use (5) in the algorithm, as it avoids explicit computation of the normal derivatives of y n i and p n i . We use (19) to show consistency with the exact solution y, p of (3). To this end, let us assume convergence.…”

“…Applications relevant for the material of this article concern the flow of gas in pipe networks (see the website (accessed on 10 January 2024) https://www.trr154.fau.de/trr-154-en/ (accessed on 10 January 2024) and the the large number of publications therein) or water networks (see, e.g., [15]). We refer readers to [16,17] and the works by Mophou et al [18,19] for more such problems involving fractional derivatives.…”