  In this work, we consider the problem of regret minimization in adaptive
minimum variance and linear quadratic control problems. Regret minimization has
been extensively studied in the literature for both types of adaptive control
problems. Most of these works give results of the optimal rate of the regret in
the asymptotic regime. In the minimum variance case, the optimal asymptotic
rate for the regret is $\log(T)$ which can be reached without any additional
external excitation. On the contrary, for most adaptive linear quadratic
problems, it is necessary to add an external excitation in order to get the
optimal asymptotic rate of $\sqrt{T}$. In this paper, we will actually show
from an a theoretical study, as well as, in simulations that when the control
horizon is pre-specified a lower regret can be obtained with either no external
excitation or a new exploration type termed immediate.
