Application of the iterative learning control of a non-linear MIMO wave equation.

•The control problem of a non-linear MIMO wave equation is considered.•To get the optimal control of the PDE, a repetitive process is used, and the applied method is based on the ILC.•Not like almost of the ILC algorithm, the initial condition here also has an iterative learning update law based on the tracking error and the state.•The convergence condition is proved using fundamental mathematical properties•Two ILC schemes are presented, and a small comparison is given.