Finite-Time Spatial Sampled-Data Control for Reaction–Diffusion Systems

The finite-time control is considered for reaction–diffusion systems (RDSs) under a designed spatial sampled-data controller (SSDC). Firstly, an SSDC is designed and the finite-time stability is investigated for RDSs, based on the designed controller. In terms on Lyapunov functional method and inequality techniques, a sufficient condition is obtained to ensure the finite-time stability. Then, uncertain RDSs are studied and the robust finite-time stability is considered. Under the designed SSDC, a criterion is obtained to ensure the uncertain RDSs to achieve robust finite-time stability. When there exist external disturbances, the H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\infty $$\end{document} performance of RDSs is investigated. Both distributed disturbances and boundary disturbances are considered, and sufficient conditions are obtained to guarantee the finite-time H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\infty $$\end{document} performance with the designed SSDC. Finally, examples are given to verify the effectiveness of our theoretical results.