  We analyze the Ericksen--Leslie system equipped with the Oseen--Frank energy
in three space dimensions. Recently, the author introduced the concept of
dissipative solutions. These solutions show several advantages in comparison to
the earlier introduced measure-valued solutions. In this article, we argue that
dissipative solutions can be numerically approximated by a relative simple
scheme, which fulfills the norm-restriction on the director in every step. We
introduce a semi-discrete scheme and derive an approximated version of the
relative-energy inequality for solutions of this scheme. Passing to the limit
in the semi-discretization, we attain dissipative solutions. Additionally, we
introduce an optimal control scheme, show the existence of an optimal control
and a possible approximation strategy. We prove that the cost functional is
lower semi-continuous with respect to the convergence of this approximation and
argue that an optimal control is attained in the case that there exists a
solution admitting additional regularity.

Analysis of PDEs