An Insensitizing control problem involving tangential gradient terms for a reaction-diffusion equation with dynamic boundary conditions
Abstract
In this article, we study the existence of insensitizing controls for a nonlinear reaction-diffusion equation with dynamic boundary conditions. Here, we have a partially unknown data of the system, and the problem consists in finding controls such that a specific functional is insensitive for small perturbations of the initial data. More precisely, the functional considered here depends on the norm of the state in a subset of the bulk together with the norm of the tangential gradient of the state on the boundary. This problem is equivalent to a (relaxed) null controllability problem for an optimality system of cascade type, with a zeroth-order coupling term in the bulk and a second-order coupling term on the boundary. To achieve this result, we linearize the system around the origin and analyze it by the duality approach and we prove a new Carleman estimate for the corresponding adjoint system. Then, a local null controllability result for the nonlinear system is proven by using an inverse function theorem.
Cite
@article{arxiv.2407.09882,
title = {An Insensitizing control problem involving tangential gradient terms for a reaction-diffusion equation with dynamic boundary conditions},
author = {Mauricio C. Santos and Nicolás Carreño and Roberto Morales},
journal= {arXiv preprint arXiv:2407.09882},
year = {2024}
}