I’m looking for a necessary and sufficient conditions such that a LTV control system

$dot{x}(t) = A(t) x(t) + u(t), x(0)=x_0$, for all $tgeq 0$,

$y(t) = C(t) x(t)$

satisfies the Perron-like result “For any bounded u there exists an initial condition x_0 such that both the state function x(t) and output function y(t) are bounded.”

One simple sufficient result is that the equation $dot{x}(t) = A(t) x(t)$ has an exponential dichotomy

and $C(t)$ is bounded. But it is surely not a necessary condition (caused by the boundedness of $C(t)$).

Could anyone help to find the necessary and sufficient condition? Thank you very much.