The Linear-Quadratic Regulation (LQR) problem with unknown system parameters
has been widely studied, but it has remained unclear whether O~(Tβ) regret, which is the best known dependence on time, can
be achieved almost surely. In this paper, we propose an adaptive LQR controller
with almost surely O~(Tβ) regret upper bound. The
controller features a circuit-breaking mechanism, which circumvents potential
safety breach and guarantees the convergence of the system parameter estimate,
but is shown to be triggered only finitely often and hence has negligible
effect on the asymptotic performance of the controller. The proposed controller
is also validated via simulation on Tennessee Eastman Process~(TEP), a commonly
used industrial process example