Optimal control problems of parabolic fractional sturm liouville equations in a star graph

TL;DR

This paper studies optimal control problems for parabolic fractional Sturm-Liouville equations on intervals and star graphs, establishing existence, uniqueness, regularity, and optimality conditions for solutions.

Contribution

It extends fractional Sturm-Liouville control theory to star graphs, providing new existence, uniqueness, and optimality results for these complex structures.

Findings

Existence and uniqueness of weak and very-weak solutions.

Characterization of optimal controls via Euler-Lagrange conditions.

Extension of control results to star graph structures.

Abstract

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.