Spatio‐temporal sampled‐data control for delay reaction‐diffusion systems

Abstract

AbstractThis article considers the exponential stabilization and performance for delay reaction‐diffusion systems (DRDSs), with spatial sampled‐data controller (SSDC) and spatio‐temporal sampled‐data controller (STSDC). Firstly, an SSDC is designed to stabilize the DRDSs. Using Lyapunov functional and Wirtinger's inequality technique, we obtain sufficient conditions to ensure the exponential stability of DRDSs. When there exist external disturbances, performance is considered and criteria are provided for the disturbed DRDSs, under the designed SSDC. Then, an STSDC is adopted for DRDSs. Time discrete item brings new difficulties for the analysis of the desired properties. To overcome these difficulties, a novel Lyapunov functional is constructed and Halaney's inequality is used. Moreover, the descriptor method is adopted. Under these techniques, delay‐dependent conditions are obtained to guarantee the exponential stability. The performance is also considered for the disturbed DRDSs under the designed STSDC. A Razumikhin‐type method is introduced together with the Lyapunov‐like functional. Sufficient conditions are presented to achieve performance for DRDSs with STSDC. Our theoretical results show that the spatial sampling interval does affect the desired properties, the shorter the spatial sampling interval, the easier to achieve the desired properties. Moreover, the time delay also influences the exponential stability of DRDSs, and smaller time delay is beneficial to the achievement of the stability. Finally, examples are given to illustrate the validity of results.