  We consider control-constrained linear-quadratic optimal control problems on
evolving surfaces. In order to formulate well-posed problems, we prove
existence and uniqueness of weak solutions for the state equation, in the sense
of vector-valued distributions. We then carry out and prove convergence of the
variational discretization of a distributed optimal control problem. In the
process, we investigate the convergence of a fully discrete approximation of
the state equation, and obtain optimal orders of convergence under weak
regularity assumptions. We conclude with a numerical example.
