In this paper we consider the linear quasi-periodic system \(\begin{aligned} \left\{ \begin{array}{l} {\dot{\theta }}=\omega ,\\ \dot{x}=(A+Q(\theta ))x,\\ \end{array} \right. \end{aligned}\) where \((x,\theta )\in \mathbb {R}^n\times \mathbb {T}^d\) , \(A\in g\) is a \(n\times n\) constant matrix with different eigenvalues, g is a matrix Lie subalgebra of \(gl(n,\mathbb {R})\) , \(\omega =\xi \bar{\omega }\in \mathbb {R}^d\) with \(\xi \in \mathcal {O}:=[\frac{1}{2},\frac{3}{2}].\) Letting \(s_0=(d+1)/2 \) and \(\beta =6n^2+6\tau -2\) , we prove that if \(Q:\mathbb {T}^d\rightarrow g\) belonging to Sobolev spaces \(H^{s+\beta }\) with each fixed \( s \ge s_0\) is sufficiently small in given \(H^{s_0+\beta }\) norm and \({\bar{\omega }}\) satisfies Diophantine condition, then there exists a Cantor set \({\mathcal {E}}\subset \mathcal {O}\) with almost full Lebesgue measure such that for any \(\xi \in {\mathcal {E}}\) , there exists a quasi-periodic transformation of the form \(\theta =\theta \) , \(x = e^{P(\theta )}y\) with \(P(\theta )\in H^s\) , which reduces above system into a constant system \({\dot{\theta }}=\omega ,\) \(\dot{y}=A_* y\) where \(A_*\in g\) is a constant matrix close to A. Different from classical smooth results, our result requires smallness conditions only on a fixed low Sobolev norm ( \(H^{s_0+\beta }\) -norm) of the first perturbation. It is worth mentioning that our system does not need second Melnikov’s condition explicitly. As an application, we apply our results to smooth quasi-periodic Schrödinger equations to study the Lyapunov stability of the equilibrium and the existenc of quasi-periodic solutions. The result can be regarded as the generalization of the stability result in [37] to the smooth category.
