This note addresses input-to-state stability (ISS) properties with respect to
(w.r.t.) boundary and in-domain disturbances for Burgers' equation. The
developed approach is a combination of the method of De~Giorgi iteration and
the technique of Lyapunov functionals by adequately splitting the original
problem into two subsystems. The ISS properties in L2-norm for Burgers'
equation have been established using this method. Moreover, as an application
of De~Giorgi iteration, ISS in L∞-norm w.r.t. in-domain disturbances
and actuation errors in boundary feedback control for a 1-D {linear}
{unstable reaction-diffusion equation} have also been established. It is the
first time that the method of De~Giorgi iteration is introduced in the ISS
theory for infinite dimensional systems, and the developed method can be
generalized for tackling some problems on multidimensional spatial domains and
to a wider class of nonlinear {partial differential equations (PDEs)Comment: This paper has been accepted for publication by IEEE Trans. on
Automatic Control, and is available at
http://dx.doi.org/10.1109/TAC.2018.2880160. arXiv admin note: substantial
text overlap with arXiv:1710.0991