This paper analyses the concept of a Limit Set in Parameter Optimal Iterative Learning Control (ILC). We investigate the existence of stable and unstable parts of Limit Set and demonstrates that they will often exist in practice. This is illustrated via a 2-dimensional example where the convergence of the learning algorithm is analyzed from the error’s dynamic behaviour. These ideas are extended to the AT-dimensional cases by analogy and example. This paper analyzes the concept of a Limit Set in Parameter Optimal Iterative Learning Control (ILC). We investigate the existence of stable and unstable parts of Limit Set and demonstrates that they will often exist in practice. This is illustrated via a 2-dimensional example where the convergence of the learning algorithm is analyzed from the error’s dynamic behavior. These ideas are extended to the AT-dimensional cases by analogy and example.