Insensitizing the tangential gradient for reaction–diffusion equations with dynamic boundary conditions

Roberto Morales (), Maurício C. Santos () and Nicolás Carreño ()
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Roberto Morales: Universidad de Sevilla
Maurício C. Santos: Federal University of Paraíba
Nicolás Carreño: Universidad Técnica Federico Santa María

Journal of Optimization Theory and Applications, 2026, vol. 208, issue 1, No 55, 35 pages

Abstract: Abstract In this article, we study the existence of insensitizing controls for a nonlinear reaction-diffusion equation with dynamic boundary conditions. The problem consists in finding a control for a system with partially unknown initial data, such that a specific functional is insensitive to small perturbations of this 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 using an inverse function theorem.

Keywords: Controllability; Heat equation; Dynamic boundary conditions; 93B05; 35Q56; 93B07 (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/s10957-025-02881-4

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