Optimal control of fractional Sturm–Liouville wave equations on a star graph

In the present paper, we are concerned with a fractional wave equation of Sturm–Liouville type in general star graph. We first give several existence, uniqueness and regularity results weak solutions for one-dimensional case using spectral theory; prove existence to quadratic boundary optimal control problem provide characterization via Euler–Lagrange first-order optimality conditions. then investigate analogous problems graph mixed Dirichlet Neumann conditions controls velocity. show minimizers, by Lagrange multipliers, able characterize controls.