论文标题
抛物线PDES空间SUP-NORM中SUP-NORM和输入状态稳定性的相对稳定性
Relative Stability in the Sup-norm and Input-to-state Stability in the Spatial Sup-norm for Parabolic PDEs
论文作者
论文摘要
在本文中，我们介绍了相对$ \ MATHCAL {K} $ - Equi-styability（RKE）的概念，以表征（弱）解决方案对非线性抛物线PDE系统外部干扰的均匀依赖性。基于RKE，我们证明了一类带有Dirichlet或Robin边界干扰的非线性抛物线PDE的空间SUP-NORM中的输入到状态稳定性（ISS）。提供了两个示例，分别与具有罗宾边界条件的超级线性抛物线PDE和具有不稳定术语的$ 1 $ -D抛物线PDE有关，以说明获得的ISS结果。此外，作为RKE的概念的应用，我们对在域上或域的边界上，空间和时间sup-norm以及空间sup-norm中分别对级联的一类PDE进行了稳定分析。 De Giorgi迭代的技术被广泛用于本文介绍的结果证明。
In this paper, we introduce the notion of relative $\mathcal{K}$-equi-stability (RKES) to characterize the uniformly continuous dependence of (weak) solutions on external disturbances for nonlinear parabolic PDE systems. 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. Two examples, concerned respectively with a super-linear parabolic PDE with Robin boundary condition and a $1$-D parabolic PDE with a destabilizing term, are 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 paper.
