Traveling profiles and control cost for a PDE describing the evolution of invasive species

We develop a detailed analysis of optimal traveling waves $U(t,x)=U(x-\beta t)$ for a model of invasive-species control proposed in [Bressan, Chiri, and Salehi, Math. Models Methods Appl. Sci., 2022] : the relative density $U\in [0,1]$ of the invasive species satisfies the following reaction-diffusion equation with a positive control \begin{equation} \label{Equa:PDE_abstract} U_t = U_{xx} + f(U) - \tilde α(t,x) U, \quad U \in [0,1], \ \tilde α\geq 0. \end{equation} The control $\tilde{\alpha }(t,x)$ represents the fraction of the population removed at $(t,x)$: the minimal control effort $E(\beta ,f)$ required to sustain a traveling invasion front with prescribed speed $\beta$ is defined as the minimal $L^1$-norm of $\tilde{\alpha }$ for a traveling wave solution $U(x-\beta t)$ to the PDE. In order to study large scale dynamics $(t,x)\mapsto (\epsilon t,\epsilon x)$, a fundamental role is played by the structure of traveling waves and the convexity and regularity properties of $E$. The main results of this paper are the following: 1)In the phase plane $(U,P=U_x)$, there exists a unique optimal profile $P_{\beta }(U)$ minimizing the effort; 2) It satisfies explicit first-order conditions, which are both necessary and sufficient; 3) The associated control is acting on an open subset of the set $\{U:P_{\beta }(U)=\sqrt{Uf(U)}\}$, in particular it is uniformly integrable, and it depends smoothly on $(\beta ,f)$ on a dense open set; 4)The effort function $E(\beta ,f)$ is only $C^1$ w.r.t. $\beta$ and Lipschitz w.r.t. $f$ in the $C^2$-topology, and is asymptocally linear for $\beta \to \mathrm{\infty }$; 5)$\beta \mapsto E(\beta ,f)$ is in general neither convex nor subadditive.