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This article investigates the dynamic behaviors of delayed fractional-order memristive fuzzy cellular neural networks via the Lyapunov method. To address the delay terms of fractional-order systems, a novel lemma is provided to make the solutions of the systems exponentially stable. Furthermore, two new intermittent-hold controllers are designed to improve the robustness of the system and reduce the cost of the controller. One intermittent-hold controller is based on the feedback control strategy, while the other one integrates an adaptive control strategy. Moreover, two crucial theorems are derived from the proposed lemma and controllers, guaranteeing the exponential synchronization between drive and response systems. Finally, the superior performance of the controllers in achieving exponential synchronization is demonstrated through simulations.