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) propertieswith respect to (w.r.t.) boundary and in-domain disturbances for Burgers'equation. The developed approach is a combination of the methodof De Giorgi iteration and the technique of Lyapunov functionals byadequately splitting the original problem into two subsystems. The ISSproperties in L2-norm for Burgers' equation have been established usingthis method. Moreover, as an application of De Giorgi iteration, ISS inL1-norm w.r.t. in-domain disturbances and actuation errors in boundaryfeedback control for a 1-D linear unstable reaction-diffusion equationhave also been established. It is the first time that the method of De Giorgiiteration is introduced in the ISS theory for infinite dimensional systems,and the developed method can be generalized for tackling some problemson multidimensional spatial domains and to a wider class of nonlinearpartial differential equations (PDEs).

Uncontrolled Keywords

De Giorgi iteration, boundary disturbance, indomain disturbance, Burgers’ equation, unstable reaction-diffusion equation.