2020/04/24 by Víctor Hernández-Santamaría, Hernández-Santamaría, Víctor, Kévin Le Balc’h +1
Engineering · Computer Science · Mathematics · #Stability and Controllability of Differential Equations #Advanced Mathematical Modeling in Engineering #Mathematical Biology Tumor Growth
paper · pdf · doi:10.48550/arxiv.2004.11923
In this paper, we prove the local-controllability to positive constant\ntrajectories of a nonlinear system of two coupled ODE equations, posed in the\none-dimensional spatial setting, with nonlocal spatial nonlinearites, and using\nonly one localized control with a moving support. The model we deal with is\nderived from the well-known nonlinear reaction-diffusion Gray-Scott model when\nthe diffusion coefficient of the first chemical species du tends to 0 and\nthe diffusion coefficient of the second chemical species dv tends to +\n\∞. The strategy of the proof consists in two main steps. First, we\nestablish the local-controllability of the reaction-diffusion ODE-PDE derived\nfrom the Gray-Scott model taking du=0, and uniformly with respect to the\ndiffusion parameter dv \∈ (1, +\∞). In order to do this, we prove the\n(uniform) null-controllability of the linearized system thanks to an\nobservability estimate obtained through adapted Carleman estimates for ODE-PDE.\nTo pass to the nonlinear system, we use a precise inverse mapping argument and,\nsecondly, we apply the shadow limit dv \→ + \∞ to reduce to\nthe initial system.\n