  We present a constructive approach to bounded $\ell_2$-gain adaptive control
with noisy measurements for linear time-invariant scalar systems with uncertain
parameters belonging to a finite set. The gain bound refers to the closed-loop
system, including the learning procedure. The approach is based on forward
dynamic programming to construct a finite-dimensional information state
consisting of $\mathcal H_\infty$-observers paired with a recursively computed
performance metric. We do not assume prior knowledge of a stabilizing
controller.
