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

TLDR: In this paper, the existence, uniqueness and regularity results of weak and very-weak solutions of a parabolic fractional initial-boundary value problem of Sturm Liouville type in an interval and in a general star graph were given.

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.