Invariant manifolds and transversality for a traveling-wave ODE in a controlled reaction-diffusion model
Abstract
We study the parameter dependence and transversality properties of trajectories of the controlled ODE Ux = P, Px = − f(U) − βP + α̃(x)U, arising from traveling-wave solutions of a bistable reaction-diffusion equation with control. Optimal profiles minimizing the L1-norm of the control α̃ are determined by the unstable and stable manifolds Γu, Γs, the critical curve P*(U) = √(Uf(U)), and the associated function F(U) = dP*/dU + f(U)/P*(U). We establish generic transversality properties for these objects with respect to perturbations of the speed β 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 β**, where the unstable manifold approaches the strongly stable manifold of the equilibrium (U*,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]].
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BibTeX
@misc{bianchini2026invariant,
title = {{Invariant manifolds and transversality for a traveling-wave ODE in a controlled reaction-diffusion model}},
author = {Stefano Bianchini and Chiara Trifone},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12105},
eprint = {2610.12105},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12105},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Invariant manifolds and transversality for a traveling-wave ODE in a controlled reaction-diffusion model
AU - Stefano Bianchini
AU - Chiara Trifone
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12105
UR - https://arxiv.org/abs/2610.12105
AB - We study the parameter dependence and transversality properties of trajectories of the controlled ODE U_x = P, P_x = − f(U) − βP + α̃(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 α̃ are determined by the unstable and stable manifolds Γ_u, Γ_s, the critical curve P^*(U) = √(Uf(U)), and the associated function F(U) = dP^*/dU + f(U)/P^*(U). We establish generic transversality properties for these objects with respect to perturbations of the speed β 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 β^**, where the unstable manifold approaches the strongly stable manifold of the equilibrium (U^*,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]].
ER -