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.