Observer-based boundary control for distributed parameter system under denial-of-service attacks

Abstract

This paper focuses on the problem of boundary control for a distributed parameter system (DPS) under denial-of-service (DoS) attacks. Initially, a DPS model is employed. Considering the incomplete measurement of the DPS's state, a novel boundary observer is then proposed, which only relies on the right boundary state instead of full-domain information to achieve accurate state estimation, significantly reducing the measurement cost. Subsequently, an anti-DoS observer-based boundary controller is designed, which is applied only to the spatial boundary to lower actuator deployment costs while improving robustness to intermittent DoS attacks. In addition, a Lyapunov-Krasovskii functional is introduced, and the design methods for the controller and observer are derived by solving linear matrix inequalities. Finally, the feasibility of the control strategy is verified through an example.

Keywords

Distributed parameter system / DoS attacks / boundary control

References

[1] Lu G.,Zhang L.,Song S.,Song X.. Observer-based quantized control for networked interconnected PDE systems with actuator failures J. Franklin Inst. 2024 361 107208

[2] Guan Y.,Jiang B.,Chen Y.. PDE-based fault-tolerant control of swarm deployment while preserving connectedness J. Franklin Inst. 2023 360 13339 58

[3] Song X.,Zhang Q.,Song S.,Ahn C. K.. Event-triggered underactuated control for networked coupled PDE-ODE systems with singular perturbations IEEE Syst. J. 2023 17 4474 84

[4] Fan X.,Xue Y.,Zhang X.,Ma J.. Finite-time state observer for delayed reaction-diffusion genetic regulatory networks Neurocomputing 2017 227 18 28

[5] Liu Q.,Lv Z.,Zou C.,Zhang H.. Finite-time stability analysis of interacting genetic regulatory networks with spatial diffusion term Comput. Biol. Med. 2025 195 110585

[6] Wang Y.,Li H.,Yang H.. Adaptive spatial-model-based predictive control for complex distributed parameter systems Adv. Eng. Inform. 2024 59 102331

[7] Wang Z. P.,Li Q. Q.,Qiao J.,Wu H. N.,Huang T.. Fuzzy boundary sampled-data control for nonlinear parabolic DPSs IEEE Trans. Cybern. 2024 54 3565 76

[8] Wang P.,Katz R.,Fridman E.. Constructive finite-dimensional boundary control of stochastic 1D parabolic PDEs Automatica 2023 148 110793

[9] Man J.,Zeng Z.,Sheng Y.. Finite-time fuzzy boundary control for 2-D spatial nonlinear parabolic PDE systems IEEE Trans. Fuzzy Syst. 2023 31 3278 89

[10] Wang Z.,Zhang X.,Qiao J.,Wu H.,Huang T.. Synchronization of reaction–diffusion neural networks with random time-varying delay via intermittent boundary control Neurocomputing 2023 556 126645

[11] Wang X.,Wu H.,Cao J.,Li X.. Dynamic event-triggered boundary control for exponential consensus of multi-agent systems of impulsive PDEs with switching topology Inform. Sci. 2024 655 119901

[12] Zhan X.,Du X.,Wu J.,Yan H.. Performance analysis of MIMO Information time-delay systems with message queue under multiple communication parameters IEEE Trans. Syst. Man. Cybern. Syst. 2024 54 88 96

[13] Zhu G.,Xu Z.,Gao Y.,Yu Y.,Li L.. Periodic event-triggered adaptive neural output feedback tracking control of unmanned surface vehicles under replay attacks Eng. Appl. Artif. Intel. 2025 146 110237

[14] Su L.,Ye D.,Zhao X.. Static output feedback secure control for cyber-physical systems based on multisensor scheme against replay attacks Int. J. Robust Nonlinear Control 2020 30 8313 26

[15] Li Y.,Liu H.,Wang H.,Lai Y.. Active security control for helicopter systems under targeted deception attacks based on Stackelberg game J. Franklin Inst. 2026 363 108300

[16] Dai H.,Guo X.,Ji L.,Li H.. Bipartite synchronization of multi-agent systems under deception attacks via pinning delayed-impulsive control Int. J. Robust Nonlinear Control 2023 33 7718 34

[17] Zhang S.,Yin X.,Gao Z.. Prescribed-time consensus control of nonlinear time-delayed multi-agent systems under DoS attacks ISA Trans. 2025

[18] Wu X.,Xu N.,Ding S.,Zhao X.,Niu B.,Wang W.. Event-based adaptive neural resilient formation control for MIMO nonlinear MASs under actuator saturation and denial-of-service attacks Inform. Sci. 2025 691 121619

[19] Wang S.,Wu D.,You Z.,Feng N.,Mao M.. Event-triggered prescribed performance control for trajectory tracking of unmanned surface vehicle under DoS attacks Ocean Eng. 2026 343 123451

[20] Wang Z.,Chen B.,Qiao J.,Wu H.,Huang T.. Fuzzy boundary sampled-data control for nonlinear DPSs with Random time-varying delays IEEE Trans. Fuzzy Syst. 2024 32 5872 85

[21] Wan M.,Xu Y.,Wu Z.. Dynamic event-triggered resilient control of nonlinear multi-agent systems against asynchronous DoS attacks IEEE Trans. Automat. Sci. Eng. 2025 22 13633 43

[22] Liu Y.,Zhang W.,Wang J.,Liu Y.,Peng J.. Event-triggered feedback control for nonlinear parabolic distributed parameter systems with time-varying delays IEEE Trans. Automat. Sci. Eng. 2025 22 5308 21

[23] Cheng J.,Tang T.,Yan H.,Wu Z.,Zhang D.. A novel switching rule to observer-based control for switched systems Automatica 2026 183 112622

[24] Farbood M.,Echreshavi Z.,Mobayen S.,Skruch P.. Resilient event-triggered observer-based control of uncertain nonlinear fuzzy systems subject to unknown inputs IEEE Access 2025 13 156359 74

[25] Wang Y.,Zeng Z.,Liu X.,Liu Z.. Input-to-state stability of switched linear systems with unstabilizable modes under DoS attacks Automatica 2022 146 110607

[26] Zhao N.,Zhang H.,Shi P.. Observer-based sampled-data adaptive tracking control for heterogeneous nonlinear multi-agent systems under denial-of-service attacks IEEE Trans. Automat. Sci. Eng. 2025 22 4771 9

[27] Wang Z. P.,Zhang X.,Wu H. N.,Huang T.. Fuzzy boundary control for nonlinear delayed DPSs under boundary measurements IEEE Trans. Cybern. 2023 53 1547 56