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Fixed-time synchronization for two-dimensional coupled reaction-diffusion complex networks: Boundary conditions

Yishu Wang1, Jianquan Lu2, Tingwen Huang3

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This study establishes fixed-time synchronization (FxTS) criteria for complex reaction-diffusion networks with impulses and delays. Proposed boundary controllers ensure rapid synchronization, validated by numerical examples.

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Area of Science:

  • Complex Networks
  • Reaction-Diffusion Systems
  • Control Theory

Background:

  • Coupled reaction-diffusion complex networks (CRDCNs) are crucial in modeling spatio-temporal phenomena.
  • Achieving synchronization in these networks, especially under impulsive and delayed conditions, presents significant challenges.
  • Fixed-time synchronization (FxTS) offers a desirable finite-time convergence without the singularity issues of finite-time control.

Purpose of the Study:

  • To investigate and establish criteria for fixed-time synchronization (FxTS) in two-dimensional coupled reaction-diffusion complex networks (CRDCNs).
  • To design effective boundary controllers that guarantee FxTS for CRDCNs subject to impulses and time delays.
  • To analyze the impact of different boundary conditions on the synchronization process.

Main Methods:

  • Lyapunov method for establishing FxTS criteria for impulsive delayed CRDCNs.
  • Lyapunov-Krasovskii functional approach for designing FxTS boundary controllers.
  • Analysis of Neumann, mixed, and vanishing Dirichlet boundary conditions.
  • Calculation of the optimal constant for the Poincaré inequality in a square domain.

Main Results:

  • A novel FxTS criterion is derived for CRDCNs with both synchronizing and desynchronizing impulses and time delays.
  • Two FxTS boundary controllers are successfully formulated for Neumann and mixed boundary conditions.
  • It is demonstrated that vanishing Dirichlet boundary conditions facilitate synchronization.
  • The optimal Poincaré inequality constant is determined for the square domain, aiding controller analysis.

Conclusions:

  • The proposed methods effectively achieve fixed-time synchronization in complex reaction-diffusion networks with challenging conditions.
  • The developed boundary controllers are robust and efficient, offering practical solutions for synchronization problems.
  • The theoretical findings are validated through comprehensive numerical simulations, confirming their practical applicability.