A TSK-type Hermite neural network (THNN) is studied in this paper. Since the output weights of the THNN use a functional-type form, it provides powerful representation, high learning performance and good generalization capability. Then, a Hermite-neural-network-based adaptive control (HNNAC) system which is composed of a neural controller and a robust compensator is proposed. The neural controller utilizes a THNN to online approximate an ideal controller, and the robust compensator is designed to eliminate the effect of the approximation error introduced by the neural controller upon the system stability. Moreover, a proportional-integral (PI)-type learning algorithm is derived to speed up the convergence of the tracking error. Finally, the proposed HNNAC system is applied to synchronize a coupled nonlinear chaotic system. In the simulation study, it shows that the proposed HNNAC system can achieve favorable synchronization performance without requiring a preliminary offline tuning.