I already have an ODE of $A(t)$, that is $\dot{A}=-G(A(t)-A^*)$, where $G$ and $A^*$ are constant positive definite matrices. Thus I can deduce that $A(t)$ exponentially converge to $A^*$.
Now I take $(A^*-A(t))$ as an input to the dynamic system $\dot{x}=-A(t)x$. By changing the system to $\dot{x}=-A^*x+(A^*-A(t))x$, hopefully $x$ can be proved to converge to $0$ by the theory of input-to-state-stability.
However, when I try to show that $f(t,x,A^*-A(t))=-A^*x+(A^*-A(t))x$ is Lipschitz in $(x,A^*-A(t))$, that is $\Vert f(t,x,A^*-A(t))-f(t,x,0) \Vert = \Vert (A^*-A(t))x \Vert \le L\Vert A^*-A(t) \Vert$, I have probelm in showing that $x$ is uniformly bounded in $t$ (that is, $\Vert x \Vert \le L$ ).
I hope that there is any theory and tools that can help me solve this problem.
Since $A(t) \to A^*$, we must have that for some $t_0$, that for $t \ge t_0$, we have $A(t)$ is positive definite (or rather, since we have no assumption on the symmetry of $A(t)$, that $A(t)+A(t)^T$ is positive definite.) Furthermore, the smallest eigenvalue of $A(t) + A(t)^T$ must be bounded below by, say, $-\lambda$, for $0 \le t \le t_0$ (because the equation for $A(t)$ is linear, and hence the solutions don't blow up in finite time).
Now take the equation $ \dot x = -A(t) x $, and dot product both sides with $x$, to get $\frac12 \frac d{dt} |x|^2 = - x^T A(t) x $. Then for $0 \le t \le t_0$, we have $\frac12 \frac d{dt} |x|^2 \le \lambda |x|^2$, and for $t \ge t_0$ we have $\frac12 \frac d{dt} |x|^2 \le 0$.
Hence $|x|^2 \le |x(0)|^2 e^{2\lambda t_0}$.