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In this paper, an adaptive neural network (NN) control with a guaranteed L infin-gain performance is proposed for a class of parabolic partial differential equation (PDE) systems with unknown nonlinearities and persistent bounded disturbances. Initially, Galerkin method is applied to the PDE system to derive a low-order ordinary differential equation (ODE) system that accurately describes the dynamics of the dominant (slow) modes of the PDE system. Subsequently, based on the low-order slow model and the Lyapunov technique, an adaptive modal feedback controller is developed such that the closed-loop slow system is semiglobally input-to-state practically stable (ISpS) with an L infin-gain performance. In the proposed control scheme, a radial basis function (RBF) NN is employed to approximate the unknown term in the derivative of the Lyapunov function due to the unknown system nonlinearities. The outcome of the adaptive L infin-gain control problem is formulated as a linear matrix inequality (LMI) problem. Moreover, by using the existing LMI optimization technique, a suboptimal controller is obtained in the sense of minimizing an upper bound of the L infin-gain, while control constraints are respected. Furthermore, it is shown that the proposed controller can ensure the semiglobal input-to-state practical stability and L infin-gain performance of the closed-loop PDE system. Finally, by applying the developed design method to the temperature profile control of a catalytic rod, the achieved simulation results show the effectiveness of the proposed controller.