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<abstract><p>The issues of exponential projective synchronization and adaptive exponential projective synchronization are analyzed for quaternion-valued memristor-based neural networks (QVMNNs) with time delays. Different from the results of existing decomposition techniques, a direct analytical approach is used to discuss the projection synchronization problem. First, in the framework of measurable selection and differential inclusion, the QVMNNs is transformed into a system with parametric uncertainty. Next, the sign function related to quaternion is introduced. Different proper control schemes are designed and several criteria for ascertaining exponential projective synchronization and adaptive exponential projective synchronization are derived based on Lyapunov theory and the properties of sign function. Furthermore, several corollaries about global projective synchronization are proposed. Finally, the reliability and validity of our results are substantiated by two numerical examples and its corresponding simulation.</p></abstract>

Summary In this paper, the global exponential synchronization of quaternion‐valued memristor‐based neural networks with time‐varying delays is discussed. Firstly, by using the differential inclusion theory and the set‐valued map theory, the discontinuous quaternion‐valued memristive neural networks is transformed into an uncertain system with interval parameters. A novel controller is designed to achieve the control goal. With the ω$$ \omega $$‐measure method and Halanay inequality, the criterion for global exponential synchronization of the quaternion‐valued memristive neural networks is given. At last, a numerical simulation is given to prove the validity of the main results.