Pinning impulsive synchronization for stochastic reaction-diffusion dynamical networks with delay.

Area of Science:

• Dynamical Systems and Control Theory
• Stochastic Processes
• Partial Differential Equations

Background:

• Stochastic reaction-diffusion networks with infinite delays present complex dynamics.
• Achieving synchronization in such systems is crucial for various applications.
• Impulsive control strategies offer a means to influence system behavior.

Purpose of the Study:

• To investigate the asymptotic synchronization in mean square of stochastic reaction-diffusion complex dynamical networks with infinite delay.
• To propose and analyze two novel pinning impulsive control strategies.
• To establish the effectiveness of these control schemes in achieving synchronization.

Main Methods:

• Utilizing the variation-of-constant formula and fixed point theorem for analyzing impulsive differential equations with infinite delay.
• Transforming the network into stochastic coupled impulsive partial differential equations in Hilbert space using abstract operators.
• Applying Lyapunov function approach and comparison principle to examine asymptotic stability in mean square.

Main Results:

• Demonstrated that asymptotic synchronization in mean square is achievable for stochastic reaction-diffusion dynamical networks under the proposed pinning impulsive control schemes.
• Showcased the efficacy of both single-node and partial-node impulsive controllers.
• Provided a theoretical framework for controlling complex network synchronization.

Conclusions:

• The proposed pinning impulsive control strategies are effective in achieving asymptotic synchronization in mean square for stochastic reaction-diffusion complex dynamical networks with infinite delay.
• The findings offer valuable insights into the control of complex dynamical systems.
• The theoretical results have potential applications, as illustrated by an example.