Fuzzy boundedness of nonlinear parabolic PDE under dynamic quantization and network attacks via dynamic output feedback

Jingzhao Chen; Liming Ding; Teng-Fei Li; Jiahuan Xue; Jingzhao Chen; Liming Ding; Teng-Fei Li; Jiahuan Xue

doi:10.3934/math.20261250

AIMS Mathematics

2026, Volume 11, Issue 9: 31739-31757. doi: 10.3934/math.20261250

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Research article

Fuzzy boundedness of nonlinear parabolic PDE under dynamic quantization and network attacks via dynamic output feedback

1.
Henan Engineering Research Center of Intelligent Manufacturing and Digital Twin, SIAS University, Zhengzhou 451150, China
2.
School of Computer and Artificial Intelligence (School of Software), Huaihua University, Huaihua 418000, China
3.
College of Aeronautical Science and Engineering, Yantai Nanshan University, Yantai 265713, China

Received: 22 July 2026 Revised: 02 September 2026 Accepted: 15 September 2026 Published: 24 September 2026
MSC : 58J35, 03B52, 39A22, 35B35

The finite-time boundedness problem was investigated for nonlinear parabolic partial differential equations (PDEs) with spatially varying coefficient matrices, using a fuzzy dynamic output feedback approach. To deal with the system nonlinearities, the Takagi-Sugeno (T-S) fuzzy model was adopted, which yielded a fuzzy parabolic PDE formulation. To reduce data transmission volume while preserving system performance, dynamic quantization was applied to both control signals and measured outputs. A spatially dependent dynamic output feedback controller was then designed to cope with network attacks. Under the imposed quantization constraints, the boundedness of the closed-loop system over a finite-time interval was established. Sufficient conditions for controller synthesis and the adjustment parameters of the quantizer were derived by means of free-matrix inequalities. In addition, a generalized matrix decomposition technique was employed to decouple the cross-coupled terms that appear in the nonlinear performance criteria. The effectiveness of the proposed method was demonstrated through simulation results on a two-dimensional dynamic model.
- parabolic PDE,
- T-S fuzzy model,
- boundedness,
- stability
Citation: Jingzhao Chen, Liming Ding, Teng-Fei Li, Jiahuan Xue. Fuzzy boundedness of nonlinear parabolic PDE under dynamic quantization and network attacks via dynamic output feedback[J]. AIMS Mathematics, 2026, 11(9): 31739-31757. doi: 10.3934/math.20261250

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Abstract

The finite-time boundedness problem was investigated for nonlinear parabolic partial differential equations (PDEs) with spatially varying coefficient matrices, using a fuzzy dynamic output feedback approach. To deal with the system nonlinearities, the Takagi-Sugeno (T-S) fuzzy model was adopted, which yielded a fuzzy parabolic PDE formulation. To reduce data transmission volume while preserving system performance, dynamic quantization was applied to both control signals and measured outputs. A spatially dependent dynamic output feedback controller was then designed to cope with network attacks. Under the imposed quantization constraints, the boundedness of the closed-loop system over a finite-time interval was established. Sufficient conditions for controller synthesis and the adjustment parameters of the quantizer were derived by means of free-matrix inequalities. In addition, a generalized matrix decomposition technique was employed to decouple the cross-coupled terms that appear in the nonlinear performance criteria. The effectiveness of the proposed method was demonstrated through simulation results on a two-dimensional dynamic model.

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