  We consider a family of controlled reaction-diffusion equations, describing
the spatial spreading of an invasive biological species. For a given
propagation speed $c\in{I\!\!R}$, we seek a control with minimum cost, which
achieves a traveling profile with speed $c$. For various nonlinear models, the
existence of a (possibly measure valued) optimal control is proved, together
with necessary conditions for optimality. In the last section we study a case
where the wave speed cannot be modified by any control with finite cost.
  The present analysis is motivated by the recent results in arXiv:2201.01723
and arXiv:2108.09321, showing how a control problem for a reaction-diffusion
equation can be approximated by a simpler problem of optimal control of a
moving set.
