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arXiv · 1811.07719

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

Jun Zheng·Guchuan Zhu
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Abstract

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 $L^\infty$-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)

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Jun Zheng, Guchuan Zhu. 2018-11-16. A De Giorgi Iteration-based Approach for the Establishment of ISS Properties for Burgers' Equation with Boundary and In-domain Disturbances. https://doi.org/10.1109/tac.2018.2880160

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