  This is a draft paper originally posted on Arxiv as a documentation of a
plenary lecture at CDC2023. The core material has been accepted for publication
at L4DC 2024. Certainty equivalence adaptive controllers are analysed using a
``data-driven Riccati equation'', corresponding to the model-free Bellman
equation used in Q-learning. The equation depends quadratically on data
correlation matrices. This makes it possible to derive simple sufficient
conditions for stability and robustness to unmodeled dynamics in adaptive
systems. The paper is concluded by short remarks on how the bounds can be used
to quantify the interplay between excitation levels and robustness.
