Spatiotemporally asynchronous sampled-data control of a linear parabolic PDE on a hypercube

Abstract

This paper employs the observer-based output feedback control technique to deal with the problem of spatiotemporally asynchronous sampled-data control for a linear parabolic PDE on a hypercube. By the spatiotemporally asynchronous sampled-data observation outputs, an observer-based output feedback control law is constructed, where the sampling interval in time is bounded. By constructing an appropriate Lyapunov–Krasovskii functional candidate and applying a weighted Poincaré–Wirtinger inequality on a hypercube, it is shown under a sufficient condition presented in terms of standard linear matrix inequalities that the suggested spatiotemporally asynchronous sampled-data control law asymptotically stabilises the PDE in the spatial ${\mathcal{H}}^1\left(\mathrm{\Omega }\right)$ norm but its convergence speed can be regulated by a known constant. Moreover, both open-loop and closed-loop well-posedness analysis are done within the framework of $C_0$ semi-group. Finally, numerical simulation results are presented to support the proposed design method.

Keywords:

• Sampled-data control
• spatiotemporally sampling asynchrony
• distributed parameter system
• observer-based feedback control
• Poincaré–Wirtinger inequality

Acknowledgments

The authors gratefully acknowledge the helpful comments and valuable suggestions of the Associate Editor and anonymous reviewers, which have improved the content and presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 A general hyperplane $[0,l_1]\times [0,l_2]\times \cdots \times [0,l_N]$ is easily revised as a hypercube $[0,1{]}^N$ by defining appropriate spatial coordinate transformation.

Additional information

Funding

This work was supported in part by Beijing Natural Science Foundation [grant number 4192037], in part by the Fundamental Research Funds for the Central Universities [grant number FRF-TP-20-07B], and in part by the National Natural Science Foundation of China (NSFC) [grant number 62073037].