  In a recent article \textit{"Lower regularity solutions of the biharmonic
Schr\"odinger equation in a quarter plane", to appear on Pacific Journal of
Mathematics [15]}, the authors gave a starting point of the study on a series
of problems concerning the initial boundary value problem and control theory of
Biharmonic NLS in some non-standard domains. In this direction, this article
deals to present answers for some questions left in [15] concerning the study
of the cubic fourth order Schr\"odinger equation in a star graph structure
$\mathcal{G}$. Precisely, consider $\mathcal{G}$ composed by $N$ edges
parameterized by half-lines $(0,+\infty)$ attached with a common vertex $\nu$.
With this structure the manuscript proposes to study the well-posedness of a
dispersive model on star graphs with three appropriated vertex conditions by
using the \textit{boundary forcing operator approach}. More precisely, we give
positive answer for the Cauchy problem in low regularity Sobolev spaces. We
have noted that this approach seems very efficient, since this allows to use
the tools of Harmonic Analysis, for instance, the Fourier restriction method,
introduced by Bourgain, while for the other known standard methods to solve
partial differential partial equations on star graphs are more complicated to
capture the dispersive smoothing effect in low regularity. The arguments
presented in this work have prospects to be applied for other nonlinear
dispersive equations in the context of star graphs with unbounded edges.
