  Adaptive control architectures often make use of Lyapunov functions to design
adaptive laws. We are specifically interested in adaptive control methods, such
as the well-known L1 adaptive architecture, which employ a parameter observer
for this purpose. In such architectures, the observation error plays a critical
role in determining analytical bounds on the tracking error as well as
robustness. In this paper, we show how the non-existence of coercive Lyapunov
operators can impact the analytical bounds, and with it the performance and the
robustness of such adaptive systems.
