Controlled Traveling Profiles for Models of Invasive Biological Species

We consider a family of controlled reaction-diffusion equations, describing the spatial spreading of an invasive biological species. For a given propagation speed $c\in I\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}R$, 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.