Relative Stability in the Sup-Norm and Input-to-State Stability in the Spatial Sup-Norm for Parabolic PDEs

• 成果类型: Article
• 署名作者: Zheng, Jun; Zhu, Guchuan; Dashkovskiy, Sergey
• 署名单位: Southwest Jiaotong University; Universite de Montreal; Polytechnique Montreal; University of Wurzburg
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2022.3192325
• 发表日期: 2022
• 页码: 5361-5375
• 关键词: Cascade of PDE systems; De Giorgi iteration; input-to-state stability (ISS); nonlinear PDEs; relative stability (RS)

摘要

In this article, we introduce the notion of relative K-equi-stability (RKES) to characterize the uniformly continuous dependence of (weak) solutions on external disturbances for nonlinear parabolic partial differential equations (PDEs). Based on the RKES, we prove the input-to-state stability (ISS) in the spatial sup-norm for a class of nonlinear parabolic PDEs with either Dirichlet or Robin boundary disturbances. An example concerned with a superlinear parabolic PDE with Robin boundary condition is provided to illustrate the obtained ISS results. Besides, as an application of the notion of RKES, we conduct stability analysis for a class of parabolic PDEs in cascade coupled over the domain or on the boundary of the domain, in the spatial and time sup-norm, and in the spatial sup-norm, respectively. The technique of De Giorgi iteration is extensively used in the proof of the results presented in this article.