  We complete the first step towards the resolution of several decades-old
challenges in disturbance-robust adaptive control. For a scalar linear system
with an unknown parameter for which no a priori bound is given, with a
disturbance that is of unlimited magnitude and possibly persistent (not square
integrable), and without a persistency of excitation necessarily verified by
the state, we consider the problems of (practical) gain assignment relative to
the disturbance. We provide a solution to these heretofore unsolved feedback
design problems with the aid of infinite-dimensional nonlinear feedback
employing distributed delay of the state and input itself. Specifically, in
addition to (0) the global boundedness of the infinite-dimensional state of the
closed-loop system when the disturbance is present, we establish (1) practical
input-to-output stability with assignable asymptotic gain from the disturbance
to the plant state; (2) assignable exponential convergence rate; and (3)
assignable radius of the residual set. The accompanying identifier in our
adaptive controller guarantees (4) boundedness of the parameter estimate even
when disturbances are present; (5) an ultimate estimation error which is
proportional to the magnitude of the disturbance with assignable gain when
there exists sufficient excitation of the state; and (6) exact parameter
estimation in finite-time when the disturbance is absent and there is
sufficient excitation. Among our results, one reveals a tradeoff between
"learning capacity" and "disturbance robustness:" the less sensitive the
identifier is to the disturbance, the less likely it is to learn the parameter.
