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

Controlled Traveling Profiles for Models of Invasive Biological Species