  An adaptive controller with bounded l2-gain from disturbances to errors is
derived for linear time-invariant systems with uncertain parameters restricted
to a finite set. The gain bound refers to the closed loop system, including the
non-linear learning procedure. As a result, robustness to unmodelled dynamics
(possibly nonlinear and infinite-dimensional) follows from the small gain
theorem. The approach is based on a new zero-sum dynamic game formulation,
which optimizes the trade-off between exploration and exploitation. An explicit
upper bound on the optimal value function is stated in terms of semi-definite
programming and a corresponding simple formula for an adaptive controller
achieving the upper bound is given. Once the uncertain parameters have been
sufficiently estimated, the controller behaves like standard H-infinity optimal
control.
