$ H_{\infty} $ boundary control design of spatial 2-D linear stochastic parabolic PDE systems

Qian Wang; Zi-Peng Wang; Zhiyi Lu; Xiao-Wei Zhang; Qian Wang; Zi-Peng Wang; Zhiyi Lu; Xiao-Wei Zhang

doi:10.3934/math.2026518

AIMS Mathematics

2026, Volume 11, Issue 5: 12606-12620. doi: 10.3934/math.2026518

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Research article

$ H_{\infty} $ boundary control design of spatial 2-D linear stochastic parabolic PDE systems

1.
School of Science, Tianjin University of Commerce, Tianjin 300134, China
2.
School of Information Science and Technology, Beijing University of Technology, Beijing 100124, China

Received: 09 March 2026 Revised: 24 April 2026 Accepted: 28 April 2026 Published: 07 May 2026
MSC : 93D15, 93C20, 35K57

In practical applications involving spatiotemporal processes, the two-dimensional (2-D) spatial scenario is often more relevant. However, higher spatial dimensionality poses a major challenge for control design, drastically increasing its complexity. This work investigated the $ H_{\infty} $ boundary control problem for a class of 2-D linear stochastic parabolic partial differential equation (PDE) systems subject to multiplicative noise in the state. We used multidimensional Green's formula and alternative inequalities to handle complex terms in 2-D systems. A static output feedback (SOF) control strategy was introduced to achieve stabilization within a stochastic paradigm. By employing Lyapunov-based technique, a constructive method for SOF controller design was established, ensuring that the closed-loop system achieved exponential stability in the mean square sense with $ H_{\infty} $ performance. Numerical simulations of a stochastic thermal process demonstrated the effectiveness and applicability of the proposed control methodology.
- 2-D linear stochastic parabolic PDE systems,
- $ H_{\infty} $ boundary control,
- static output feedback,
- mean square exponential stability
Citation: Qian Wang, Zi-Peng Wang, Zhiyi Lu, Xiao-Wei Zhang. $ H_{\infty} $ boundary control design of spatial 2-D linear stochastic parabolic PDE systems[J]. AIMS Mathematics, 2026, 11(5): 12606-12620. doi: 10.3934/math.2026518

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Abstract

In practical applications involving spatiotemporal processes, the two-dimensional (2-D) spatial scenario is often more relevant. However, higher spatial dimensionality poses a major challenge for control design, drastically increasing its complexity. This work investigated the $ H_{\infty} $ boundary control problem for a class of 2-D linear stochastic parabolic partial differential equation (PDE) systems subject to multiplicative noise in the state. We used multidimensional Green's formula and alternative inequalities to handle complex terms in 2-D systems. A static output feedback (SOF) control strategy was introduced to achieve stabilization within a stochastic paradigm. By employing Lyapunov-based technique, a constructive method for SOF controller design was established, ensuring that the closed-loop system achieved exponential stability in the mean square sense with $ H_{\infty} $ performance. Numerical simulations of a stochastic thermal process demonstrated the effectiveness and applicability of the proposed control methodology.

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