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A new framework to design immersion and invariance adaptive controllers for nonlinearly parameterized, nonlinear systems was recently proposed by the authors. The key step is the construction of a monotone mapping, via a suitable selection of a controller tuning function, which has to satisfy some integrability conditions-this translates into the need to solve a partial differential equation (PDE). In this paper this result is extended providing some answers to the questions of characterizationdoi:10.3182/20110828-6-it-1002.01708 fatcat:2lkojtwzyvdznh3pdkgytmlk74