  In this paper, we are interested in the relation between the solutions of the
control system $\dot x=f(x,u)$ and the solutions of its (potentially unknown)
perturbation $\dot x=f(x,u)+w(x,t).$ Under the assumption that the linear part
of the unperturbed system at the point $(0,0)$ is controllable and that
disturbance $w(x,t)$ is asymptotically sufficiently small, there exists a
state-feedback controller of the form $u=-Kx$ such that the perturbed system
preserves the local asymptotic stability of the zero solution of unperturbed
system. The main result of this paper gives the sufficient conditions, more
specifically, the relations between the important parameters of the system, to
ensure this property and at the same time provides the method for calculating
the lower bound of region of attraction. Moreover, we obtain a nontrivial
extension of the classical result of H. K. Khalil regarding asymptotic behavior
of the (uncontrolled) perturbed systems whose nominal part is exponentially
asymptotically stable at the origin $x=0.$
