Invariant manifolds and transversality for a traveling-wave ODE in a controlled reaction-diffusion model

We study the parameter dependence and transversality properties of trajectories of the controlled ODE $$U_x=P,P_x=-f(U)-\beta P+\tilde{\alpha }(x)U,$$ arising from traveling-wave solutions of a bistable reaction-diffusion equation with control. Optimal profiles minimizing the $L^1$-norm of the control $\tilde{\alpha }$ are determined by the unstable and stable manifolds ${\mathrm{\Gamma }}_u$, ${\mathrm{\Gamma }}_s$, the critical curve $$P^{\ast }(U)=\sqrt{Uf(U)},$$ and the associated function $$F(U)=\frac{d{P}^{\ast }}{dU}+\frac{f(U)}{P^{\ast }(U)}\mathrm{.}$$ We establish generic transversality properties for these objects with respect to perturbations of the speed $\beta$ and the nonlinear source $f$, together with quantitative estimates on the first- and second-order dependence of the invariant manifolds on these parameters. We also analyze the loss of regularity at the critical speed ${\beta }^{\ast \ast }$, where the unstable manifold approaches the strongly stable manifold of the equilibrium $(U^{\ast },0)$. These results provide the ODE ingredients used in the analysis of the structure and regularity of optimal profiles in paper [S. Bianchini and C. Trifone. Traveling profiles and control cost for a PDE describing the evolution of invasive species. 2026. arXiv: 2606.23979 [math.OC]].