For this, we derive the weighted estimates for discreet Green function and the asymptotic error expansion inequalities, and then the proofs of the formulas are given.

A periodic mixed adaptive law and an adaptive repetitive learning control one are constructed to ensure the asymptotic convergence of the tracking error in the sense of square error norm.

Our proof of the convergence is based on an asymptotic diffusion expansion and requires error estimates on a matched boundary layer approximation to the solution of the discrete-ordinate method.