A Local Result on Insensitizing Controls for a Semilinear Heat Equation with Nonlinear Boundary Fourier Conditions
Olivier Bodart, Manuel González-Burgos, Rosario Pérez-García · SIAM Journal on Control and Optimization · 2004
In this paper we present a local result on the existence of insensitizing controls for a semilinear heat equation when nonlinear boundary conditions of the form $\partial_n y + f(y) = 0$ are considered. The problem leads to an analysis of a special type of nonlinear null controllability problem. A sharp study of the linear case and a later application of an appropriate fixed point argument constitute the scheme of the proof of the main result. The boundary conditions we are dealing with lead us to seek a fixed point, and thus also control functions, in certain Hölder spaces. The main strategy in this paper is the construction of controls with Hölderian regularity starting from L 2 -controls in the linear case. Sufficient regularity in the data and appropriate assumptions on the right-hand side term $\xi$ of the equation are required.