  Recent advancements in adaptive control have equipped deep neural network
(DNN)-based controllers with Lyapunov-based adaptation laws that work across a
range of DNN architectures to uniquely enable online learning. However, the
adaptation laws are based on tracking error, and offer convergence guarantees
on only the tracking error without providing conclusions on the parameter
estimation performance. Motivated to provide guarantees on the DNN parameter
estimation performance, this paper provides the first result on composite
adaptation for adaptive Lyapunov-based DNN controllers, which uses the Jacobian
of the DNN and a prediction error of the dynamics that is computed using a
novel method involving an observer of the dynamics. A Lyapunov-based stability
analysis is performed which guarantees the tracking, observer, and parameter
estimation errors are uniformly ultimately bounded (UUB), with stronger
performance guarantees when the DNN's Jacobian satisfies the persistence of
excitation (PE) condition. Comparative simulation results demonstrate a
significant performance improvement with the developed composite adaptive
Lb-DNN controller in comparison to the tracking error-based Lb-DNN.
