Stabilization of a reaction-diffusion equation in H2-norm with application to saturated Neumann measurement

This paper is concerned with the boundary output stabilization of a reaction-diffusion equation in $H^{2}$-norm. Stabilizability of reaction-diffusion PDEs is most often studied in $L^{2}$-norm and sometimes in $H^{1}$-norm. The case of the $H^{2}$-norm is much less reported in the literature. In this paper, the study of the system trajectory in $H^{2}$-norm is motivated by the fact that such a regularity is required, from a mathematical perspective, to handle a saturated Neumann measurement. More precisely, combining a classical sector condition for saturation functions and spectral methods, we show how the study of the system in $H^{2}$-norm allows the local exponential stabilization of the PDE plant while estimating a subset of the domain of attraction.