# stability – Perron – like theorem for LTV systems & BIBO stable

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.