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Finite dimensional adaptive H∞ control for distributed parameter systems of hyperbolic type preceded by input nonlinearities such as dead-zone or backlash, is considered in this paper. Distributed parameter systems are infinite dimensional processes, but the proposed control scheme is constructed from finite dimensional controllers. An adaptive inverse model is introduced to estimate and compensate the input nonlinearity. The stabilizing control signal is added to regulate the effect of spill-over terms, and it is derived as a solution of certain H∞ control problem where the residual part of the inverse model and the spill-over term are regarded as external disturbances to the process.