Stability of linear time varying system

Question

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.

Answer 1

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 $e^{-kt}$, $k>0$.