A De Giorgi Iteration-Based Approach for the Establishment of ISS Properties for Burgers' Equation With Boundary and In-domain Disturbances

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 L-2-norm for Burgers' equation have been established using this method. Moreover, as an application of De Giorgi iteration, ISS in L-infinity-norm w.r.t. in-domain disturbances and actuation errors in boundary feedback control for a one-dimensional 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.