On the local asymptotic stabilization of the nonlinear systems with small time-varying perturbations by state-feedback control

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 $x=0.$