Consider the following time-varying homogeneous system $\dot{x}(t)=A(t)x(t)$. Assume that there exist an $n\times n$ symmetric matrix $Q(t)$ that satisfies $$\nu I\leq Q(t)\leq \rho I,$$ $$A^T(t)Q(t)+Q(t)A(t)+\dot{Q}(t)\leq 0,$$ where $\nu,\rho$ are positive constants. Show that given system is uniformly stable under this condition.
Some things that I found might be useful:
Definition: $x_e$ is uniformly stable iff for $\varepsilon>0,\exists\delta=\delta(\varepsilon)>0$ s.t. $\left\|x_0-x_e\right\|\leq\delta \Rightarrow \left\|x(t;t_0,x_0)-x_e\right\|\leq\varepsilon,\forall t\geq t_0.$
Theorem: All eigenvalues of $A$ have negative real parts ($\dot{x}(t)=A(t)x(t)$ is stable) iff for any given positive definite symmetry matrix $N$, the Lyapunov equation $$A^TM+MA=-N$$ has a unique symmetric solution $M$ and $M$ is positive definite.