Gradient extremum seeking for compensating wave actuator dynamics in cascade with static scalar maps is addressed in the present paper. This class of Partial Differential Equations (PDEs) for extremum seeking has not been studied yet. A dynamic feedback control law based on distributed parameters is proposed by employing backstepping transformation with an appropriate target system and an adequate formulation using Neumann interconnections. Local stability and convergence to a small neighborhood of the desired (but unknown) extremum is proved by means of a Lyapunov functional and the theory of averaging in infinite dimensions. Numerical simulations illustrate the theoretical results.
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