Controlled traveling profiles for models of invasive biological species

Abstract

We consider a family of controlled reaction-diffusion equations, describing the spatial spreading of an invasive biological species. For a given propagation speed c ∈ IR, we seek a control with minimum cost, which achieves a traveling profile with speed c. Since our goal is to slow down or even reverse the contamination, we always assume c > c*, where c* is the speed of an uncontrolled traveling profile. For various nonlinear models, the existence of an 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 A. Bressan, et al. Math. Models Methods Appl. Sci. 32 (2022) 1109–1140. and A. Bressan, et al. J. Differ. Equ. 361 (2023) 97–137, showing how a control problem for a reaction-diffusion equation can be approximated by a simpler problem of optimal control of a moving set.