Observer-based boundary stabilization of a reaction–diffusion equation under random measurement deception: a dynamic event-triggered design

Lianglin Xiong; Xueqing Chen; Haiyang Zhang; Tao Wu; Lianglin Xiong; Xueqing Chen; Haiyang Zhang; Tao Wu

doi:10.3934/math.20261117

AIMS Mathematics

2026, Volume 11, Issue 9: 27969-28008. doi: 10.3934/math.20261117

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Observer-based boundary stabilization of a reaction–diffusion equation under random measurement deception: a dynamic event-triggered design

1.
School of Mathematics and Computer Science, Yunnan Minzu University, Kunming 650504, Yunnan, China
2.
Yunnan Open University, Kunming 650504, Yunnan, China
3.
School of Mathematics and Statistics, Yunnan University, Kunming 650504, Yunnan, China

Received: 10 April 2026 Revised: 21 August 2026 Accepted: 25 August 2026 Published: 02 September 2026
MSC : 35K57, 93C20, 93D15

This work considers dynamic event-triggered boundary output feedback control for a one-dimensional reaction–diffusion equation when the boundary measurement may be randomly falsified. The compromised measurement channel is described by a Bernoulli switching model together with a Lipschitz nonlinear deception mapping. To alleviate the control update and actuation burden in networked implementation, a dynamic triggering law equipped with an auxiliary internal state is introduced so that the triggering threshold is adjusted online according to the closed-loop evolution. It is further shown that the triggering instants admit a uniform positive separation, thereby excluding Zeno accumulation. Based on this triggering framework, a backstepping boundary observer and its corresponding boundary feedback law are constructed. A composite Lyapunov analysis is then carried out to establish sufficient conditions for a global mean-square exponential estimate of the plant and observer states with respect to the augmented initial energy despite the coexistence of random deception and event-triggered updating. Numerical simulations, including Monte Carlo tests under independent attack realizations, are finally provided to illustrate the theoretical results. The results show that the proposed scheme stabilizes the originally unstable reaction–diffusion system, maintains effective state estimation under intermittent deceptive measurements, guarantees a positive interevent time, and substantially reduces unnecessary boundary control updates while preserving the desired closed-loop convergence behavior.
- reaction–diffusion equation,
- boundary observer,
- event-triggered boundary control,
- deceptive measurement attacks,
- global mean-square exponential stability
Citation: Lianglin Xiong, Xueqing Chen, Haiyang Zhang, Tao Wu. Observer-based boundary stabilization of a reaction–diffusion equation under random measurement deception: a dynamic event-triggered design[J]. AIMS Mathematics, 2026, 11(9): 27969-28008. doi: 10.3934/math.20261117

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Abstract

This work considers dynamic event-triggered boundary output feedback control for a one-dimensional reaction–diffusion equation when the boundary measurement may be randomly falsified. The compromised measurement channel is described by a Bernoulli switching model together with a Lipschitz nonlinear deception mapping. To alleviate the control update and actuation burden in networked implementation, a dynamic triggering law equipped with an auxiliary internal state is introduced so that the triggering threshold is adjusted online according to the closed-loop evolution. It is further shown that the triggering instants admit a uniform positive separation, thereby excluding Zeno accumulation. Based on this triggering framework, a backstepping boundary observer and its corresponding boundary feedback law are constructed. A composite Lyapunov analysis is then carried out to establish sufficient conditions for a global mean-square exponential estimate of the plant and observer states with respect to the augmented initial energy despite the coexistence of random deception and event-triggered updating. Numerical simulations, including Monte Carlo tests under independent attack realizations, are finally provided to illustrate the theoretical results. The results show that the proposed scheme stabilizes the originally unstable reaction–diffusion system, maintains effective state estimation under intermittent deceptive measurements, guarantees a positive interevent time, and substantially reduces unnecessary boundary control updates while preserving the desired closed-loop convergence behavior.

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