In order to solve the problem that the output of special repetitive operation systems, like manipulators in industrial process, only needs to track some special key points on the desired trajectory rather than realize full trajectory tracking in limited time, a norm-optimal point-to-point iterative learning control algorithm is proposed for a linear time-invariant discrete system. By transforming the matrix model of the input and output time series, a comprehensive multi-objective point performance index function is constructed. Thus, the optimal iterative learning control law can be obtained through the quadratic optimal solution. At the same time, the sufficient conditions for convergence of the robust control algorithm in the form of largest singular value are given in the case of model nominal and uncertain. Moreover, the convergence results of the optimal control algorithm for systems with input signal constraints are further generalized. Finally, the effectiveness of the algorithm is verified on the three-axis gantry robot model.