Author studies the approximation method for the optimal stabilization control of the quasilinear system \(\dot x=Ax+Du+\mu \varphi (x)\), where \(\mu\) is a small parameter, \(A\) and \(D\) are matrices, the function \(\varphi (x)\) is analytic in some bounded domain, components of the vector-valued function \(x(t,\mu)\) are absolutely continuous, components \(u_{s}(t,\mu), s=1,\dots,m\), of the vector-valued function \(u(t,\mu)\) satisfy the conditions \(|u_{s}|\leq u_{s}^{*}, s=1,\dots,m\). An algorithm for finding the \(\overline u(x,\mu)\) such that for any \(\varepsilon>0\): \(|I(x_{0},\overline u(x,\mu)-\inf_{\{u\}}I(x_{0},u)|<\varepsilon\), where \(I(x_{0},u)=\int_{0}^{\infty}[\omega^{(2)}(x)+\sum_{s=1}^{m} P_{s}(u_{s})] dt\); \(x_{0}\) is an initial data; \(\omega^{(2)}(x)\) is a positive definite quadratic form; \(P_{s}(u_{s})\geq 0, s=1,\dots,m\), is proposed. The cases a) \(P_{s}(u_{s})=0\); b) \(P_{s}(u_{s})=\delta_{s}(u_{s})^2\); c) \(P_{s}(u_{s})=\delta_{s}|u_{s}|\) are considered.