Null controllability of Chafee–Infante equation under discrete-time point measurements

Abstract

Nonlinear system is one of the main research objects in cybernetics, and it is the main theme of cybernetics in the 21st century. Recently, the control of the reaction–diffusion equation has been widely studied, but the nonlinear reaction–diffusion equation has been rarely studied. This paper will take the Chafee–Infante equation as an example, and the null controllability of this equation will be shown. We consider the null controllability for Chafee–Infante equation with point actuations subject to a known constant delay. The point measurements can be sampled in time and transmitted through a communication network with a time-varying delay. We design an observer for the future value of the state in order to compensate the input delay, then we ensure that the estimation error vanishes exponentially with a desired decay rate by using a time-varying observer gain. By constructing Lyapunov–Krasovskii functional and combining linear matrix inequalities (LIMs), we obtain the convergence conditions. We design the boundary controller and the point controller, and we conclude that both controllers can ensure the exponential stability of the closed-loop system with an arbitrary decay rate, which is smaller than that of the observers estimation error. At last, numerical example is given.