  This work presents and analyzes space-time finite element methods on fully
unstructured simplicial space-time meshes for the numerical solution of
parabolic optimal control problems. Using Babu\v{s}ka's theorem, we show
well-posedness of the first-order optimality systems for a typical model
problem with linear state equations, but without control constraints. This is
done for both continuous and discrete levels. Based on these results, we derive
discretization error estimates. Then we consider a semilinear parabolic optimal
control problem arising from the Schl\"ogl model. The associated nonlinear
optimality system is solved by Newton's method, where a linear system, that is
similar to the first-order optimality systems considered for the linear model
problems, has to be solved at each Newton step. We present various numerical
experiments including results for adaptive space-time finite element
discretizations based on residual-type error indicators. In the last two
examples, we also consider semilinear parabolic optimal control problems with
box constraints imposed on the control.
