# Stability of linear time varying system

In the case of LTV systems, of the form

$$\dot{x} = A(t)x$$

the notion of uniformly globally asymptotical stability and globally exponential stability, are they one and the same? If possible, can anyone suggest any examples?

Thank you.

No, they aren't. A counterexample is the system $$\tag{1} \dot x=-x/t,\qquad t\in[1,+\infty).$$ The solution to the initial value problem $$\dot x=-x/t,\qquad x(t_0)=x_0$$ is given by $$x(t)= x_0\cdot \frac{t_0}{t}.$$ Since $$t\ge t_0$$, we have $$\forall t\ge t_0\;\;|x(t)|\le|x_0|.$$ This means that the system (1) is uniformly stable by definition (one can take $$\delta=\epsilon$$ in the definition of uniform stability). But the system (1) is not exponentially stable because the reciprocal function decays slower than any function of the form $$e^{-kt}$$, $$k>0$$.