An Insensitizing control problem involving tangential gradient terms for a reaction-diffusion equation with dynamic boundary conditions

TL;DR

This paper investigates insensitizing controls for a nonlinear reaction-diffusion system with dynamic boundary conditions, introducing new Carleman estimates and establishing local null controllability through duality and inverse function techniques.

Contribution

It develops a novel approach to insensitizing control problems involving tangential gradient terms and dynamic boundary conditions for reaction-diffusion equations.

Findings

Established existence of insensitizing controls for the system.

Proved a new Carleman estimate for the adjoint system.

Demonstrated local null controllability of the nonlinear system.

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.