Is normalization necessary for stable model reference adaptive control?

One of the longest standing open questions in adaptive control concerns the correctness of the stability claim of the un-normalized model reference scheme proposed by Monopoli in 1974. Although provably correct solutions to the problem now abound, in particular, it is well known that adding a normalization to Monopoli´s original scheme ensures global convergence, it is interesting to know whether this technique-driven modification is really necessary or only required to complete the stability proof in the absence of more elaborate arguments. In this note, we construct a counterexample that provides a definite-unfortunately, negative-answer to the claim. Instrumental for the establishment of this result is a technical lemma that shows that, under some conditions on the regressor that may appear in Monopoli´s scheme, the parameter error freezes as the adaptation gain goes to infinity. On the lighter side, we also prove that Monopoli´s scheme is semiglobally stable, underscoring the relevance of this important contribution.